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,
sinq^shKh,
-costpjSintp,
cos
1
2
2
1
1
2
2
2
2
(77)
224 The key problem in deriving complex structure-property relations for composites with non-aligned inclusions in terms of transformation theory is the integration leading to the average S tensor s' =
!
m
Q
* ( a > W
> * * V t o V
Q
(78)
all orientations 8
Until recently, the 3 = 6561 volume integrations required to obtain s' for a given orientation distribution function have presented an obstacle to the full practical implementation of transformation theory to composites with non-aligned inclusions. Thus only relatively simple orientation distribution functions have been handled so far. Brown and Clarke have obtained average S tensors for families of inclusions on cube planes by cyclic permutation of tensor indices. Takao et a l . ' have obtained numerical solutions for uniform and cosine-type orientation distribution functions. However, the advent of computer software, capable of analytical integration by algebraic manipulation of symbols appears to have largely removed the computational problems associated with S[j To obtain an impression of the power of these computer facilities let us consider the case of a 3D random (g(Q) = 1) composite with identically shaped ellipsoidal inclusions. The average S tensor is in this case ijkl
57
73
74
U
*W
=
- f f / 0*
/ 0
0
X
X
sin< )d<
d< )d(
l imVfa. 4> f Pi l P2
rim
0
with the I- given by Eq. (77). In spite of the complexity of the 6561 triple integrals involved in Eq. (79) the programming task using the algebraic software R E D U C E turned out to be minimal. A mere 24 line programme and approximately 2 CPU hours were sufficient to obtain the complete answer: s' has 12 non-zero components and a symmetry, which can be represented by 75
ijkl
\
•
i
\
•
x x
The 3 non-identical components are
a
225 s'
- ¥'1111
S
5 1111
+
c
+- l U ll" " 15^ 3311 +
1
S
12
1122 =
S
5/ ?'
S
5
1
j .
P
p
=
S
, 2-2c + J
15
—"
- -L
is^llll
1212
c
3322
. l o" 1
+
p
_
*3333
1
S
M 3322
+
5
S
A S
7^
+
+
+ +
7J 2222 '2222 5
4
4
1
1
"
^22U
77 3333
+
+
7 1212 5
1 5
4O
4 313
5
^^2233
++
4C o _ 4C„ 7^1212 ^ 1
v
S
+ +
^ 2211
7^1133
30 1122
S
+
15
-
7 2222
+
C
I> S
+
S
j .
5
*c , 4 4 77 1313 77 7J T7 1212 1212 7J 77 1313
x
^ 1133
+
O
30" 3311 __1_
+
7*3333 , 3333
++
,i
0
1
3322 n°3322
15
+
15 3311
_
7J' 1133 7J 2211
+
7J 1111 ^ 1 1 2 2
S
2 +, —
1_S7
S
^ U22
~
^2323
c
1c 30
22
33
7^*2222" T^ : 1
+
5
| 1313
+
7^2323
and, finally, insertion o f any o f the various mathematical expressions for the components of S for continuous fibres, discontinuous fibres, disks, ribbons and spheres " leads in every case to the expected result 56
3 ~
7
=
V J
7
, <56
1 1 1 1
?' J
12
ic J
37
1 1
1122
_ =
58
7 - 5v 15(1 - v ) 5v
1
15(1 - v ) 5v
= J
1212
15(1 - v )
which is the S matrix for a spherical inclusion. In other words, the answer simply confirms Walpole's general demonstration that the average phases stresses in composites with overall elastic isotropy display a connection with Eshelby's solution for a transformed spherical inclusion. Thus in a sense one might say that nothing new has been learned: 3D random phase structures lead to simple structure-property relations for elastic composites. However, the result illustrates the promise of analytical computer methods in connection with the transformation theory for composites: the simplicity and versatility of the analytical approach should enable it to deliver simple and rigorously based descriptions of a wide range o f complex structure-property relations for real composites of engineering importance. 55
As a comparatively simple practical illustration of how a complex relation can be generated let us consider a 2D random composite o f discontinuous fibres, oriented on average parallel to the x x plane but randomly oriented within the plane. This problem was solved recently , but without the aid of REDUCE, thus involving extensive and 2
76,77
3
226 tedious computer progranuning. We assume for simplicity that the fibres have identical S matrices and lie strictly within the x x plane. The average S tensor then becomes 2
3
2n
sL with 1 0 SL = (
0
0 cos6
-sin0
0 sinG
cos0
t
In the 2D random case a 20 line REDUCE programme and about 10 CPU minutes was enough to reveal 12 non-zero components and the symmetry
• • • i
. X .
S' =
x
S
S
22
23
Here, the 7 non-identical components are
a
ll
•'mi = Sun ?' = 1^1122 1122 - s' 22n 2211 - 2222 ?' (3S -
+
S
J
*L
= \v
J
J
-
g
* 33ll) S
1-
2 2 2 2
s' ~ "^( 2222 2233 S* -(S °2323 S
+
JJ
J 1
is 2
2 « S
¥ J
1212
= {Puia
S
2233
2233
J
2233
2 2 2 2
*44
ll33)
+
^1313^
+
+
*3322
+
^^3322 *3322
+
3*3333 +
*3333
*3333
+
+
4
*2323> ^2323)
4S
2323
)
227 While S then has the symmetry of the 5 tensor for an oblate spheroid normal to the x, plane, its components, not surprisingly, turn out to depend upon the aspect ratio p of the fibres. Thus by inserting the S components for short fibres the programme finds ijkl
= QK
^21
=
+
^
{2n-\l )R -
-
=
* (7r i-/ )i? +
-1)7
2 6« S
=
*
T-Qn - (*+\l )R c
24^
^
2
r
*
c
8
23
s
1)7/ 4 -1)7/
= -Q% * (71 +\l )R J
2
a
+
4
c
4
1)7/
-Qn * ( T U l / ) J ? 6 +
7
a
2
where
g
_
3
^ _
8it(l-v) '
d-2v) 87t(l-v) '
T
_
Q(47t-3/ ) a
2
3(p -l)
In practice the alumina fibres in the squeeze-cast systems display a distribution of angular deviations from the x£x ' plane, and advanced structural characterization techniques provide experimental information on such non-random distributions. In combination with analytical computer methods the transformation theory seems an ideal theoretical basis for the exploitation of this type of experimental information in practical composite modelling. It should perhaps be noted that the averaging procedure for S tensors ignores the geometrical constraints on packing of rigid non-aligned fibres. Thus, Kelly and Parkhouse show that the maximum obtainable volume fraction of straight 2D or 3D randomly oriented rigid fibres tends to zero as 1/d becomes large. In reality the elastic flexibility of fibres allows maximum packings of / = 0.2 for 2D and 3D random fibres of large 1/d values. 3
78,79
80
228 4.2. Numerical
Modelling
The representative regions of effective medium theory are consistent with myriads of specific phase structures. In determining the effective properties, these specific geometries are all averaged out, although some of them may be critical for the initiation and evolution of damage and failure of the composite. However, information on the role of specific phase structures may be obtained using the finite element method, developed for accurate stress-strain analysis of complex full scale engineering components. In the "cell models" ' a specific array of matrix and reinforcement (the unit cell) with judiciously chosen boundary conditions is defined for subsequent meshing and numerical computation of stresses and strains. Computational requirements still limit the unit cells to fairly simple geometries, as exemplified in Fig. 14. 30
31,81,82
h/H=0
a
h/H=l/4
D D D D D Ol D D 0 D
<e> h/H=l <e> h/H=0
h/H=l
Figure 14. A selection of unit cells for numerical finite element analysis of structure-property relations for discontinuous or continuous fibre composites, (a) Systematically varied periodic arrays of continuous parallel fibres , (b) Periodic array of parallel discontinuous fibres , (c) Finite element mesh for numerical analysis of periodic arrays of non-parallel discontinuous fibres , (d) Mesh for numerical analysis of a large unit cell with a random array of 30 parallel continuous fibres . 31
81
82
30
229 Comparison o f the unit cells with the representative regions in effective medium theory suggests that the cell models w i l l in general deliver complex relations. In particular, the cell models transcend conventional effective medium theory when the elasto-plastic response of the phases is included. In practice the plastic deformation of metal matrix composites typically involves large plastic strains in metal forming operations; and large plastic strains may evolve locally* , even in service conditions where the overall plastic strain typically is small. Computationally, the large strain problem is solved by a periodic remeshing procedure, referred to as the Lagrangian convected coordinate formulation . Due to hmitations in physical understanding, computations involving large plastic strains usually still involve the uncertainty of extrapolating constitutive equations for plasticity beyond their domains of application. In comparing the numerical cell model approach with effective medium theory, it is of interest to address the concept of "best bounds"* . It is possible to construct a specific phase geometry with overall isotropy (the composite spheres assemblage ), whose effective moduli can be calculated exactly and shown to coincide with the H i l l Hashin-Shtrikman bounds. Consequently, these bounds must be the best possible, that is, the most restrictive that can be obtained in terms of phase elastic moduli and volume fractions. In principle, it should be possible to specify unit cells for every point between best bounds. This crucial issue was recently considered by Shen, Finot, Needleman and Suresh , who examined a set of unit cell models with equiaxed inclusions, as well as one 1
81
32 33
88
31
a
1
W.*
b
O.O
H-o
INCLUSION VOLUME FRACTION 31
INCLUSION VOLUME FRACTION
Figure 15. Numerical estimates compared with the analytical estimates shown in Fig. 12. (a) Dashed curve refers to unit cylinders (l/d=l) and dotted curve refers to spherical inclusions, (b) Dashed curve refers to short cylinders (l/d=5). The numerical results can be seen to support the self-consistent estimates, whether the inclusions are in the shape of particles (a) or whiskers (b).
230 unit cell model with aligned whiskers, defined as short fibres in the shape of circular cylinders of aspect ratio 1/d = 5. The Youngs moduli for their cell models with equiaxed inclusions were found to span almost a third of the distance between the upper and lower best bounds for isotropic composites with maximum and minimum predictions corresponding to unit cylinders and spheres, respectively. While the best bounds then remain to be completely spanned by the unit cell models for differently shaped inclusions, the numerical study provides support to the self-consistent estimate (SCE). As Fig. 15 shows, the cell model with spherical inclusions matches the SCE for isotropic two-phase composites very accurately for volume fractions up to about / = 0.35. Above this value the cell model predicts a slightly softer overall elastic response than does the selfconsistent model. The cell model with aligned whiskers predicts an anisotropic elastic response for the composite and it therefore deviates from the best bounds for isotropic elastic composites. However, the self-consistent model was recently evaluated by Pedersen and Withers for short ellipsoids of various aspect ratios c/a, see section 3.5. In Fig. 15b the cell model prediction can be seen lying roughly midway between the SCE and the lower bound. Due correction was made for the small difference between the modulus ratio E /E = 6.3 assumed by Shen and coworkers, and the ratio E /E = 6.9 assumed by Pedersen and Withers. It is tempting to think of the difference between the SCE for 10
F
M
F
M
2.2
/ / E R. =6.3EM
. /
•<— or
, . <— <e> J
v« = 0 . 1 7 /
-
1.2
u.OOO
0.100
0.200
0.300
0.400
Reinforcement Area Fraction 31
Figure 16. Numerical calculations of the effective Young's modulus normalized by Young's modulus of the unreinforced matrix (under plane strain (p.s.) conditions). The calculations were made for the square arrays , and <e> shown in Fig. 14. The shaded area is added by the present authors: it represents the upper and lower bound estimates of the transverse Young's modulus delivered by the variable constraint model. 29
231 ellipsoids and the unit cell prediction for cylinders as a shape effect. However, there is the apparent inconsistency that for equiaxed inclusions the unit cylinder prediction exceeds the SCE, while for whiskers the SCE exceeds the cylinder prediction. Nevertheless, the maximum disagreement between the numerical result and the SCE stays within about 5%. In general terms, the cell models and the models derived from transformation theory then all imply that inclusion shape effects arise from differences in the efficiency of load transfer from matrix to inclusions: The load transfer to equiaxed inclusions is less efficient than that to elongated inclusions, as was evident already from the simple shear lag model . Effective medium theory and the models derived from transformation theory involve the additional idea that the inclusions interact elastically with each other and with the applied stress via the spatially fluctuating field surrounding them. This feature implies the occurrence of inclusion distribution effects. Such effects are well illustrated by cell model studies of different periodic arrays of inclusions. Fig. 14 shows the quarter unit cells for periodic arrays of continuous aligned fibres with square crosssection studied by Shen et al. The configurations range systematically from square edgepacking (labelled or ), to square diagonal-packing (labelled <e>). Fig. 16 shows the corresponding variation o f the plane strain tensile stiffness as a function of fibre volume fraction for the ratio E /E = 6.3 of Youngs modulus for fibres and matrix. The inclusion distribution effects in the cell models were correlated with spatial fluctuations of hydrostatic stress in the matrix. The effects span about a third of the separation between the upper and lower bound estimates of the variable constraint model, although the comparison with estimates for transversely isotropic composites is not strictly valid, due to the the transverse anisotropy of the periodic distributions. Recent w o r k suggests that advances in effective medium theory may lead to bounds even for transversely anisotropic fibre composites. Brockenbrough, Suresh and Wienecke extend the numerical studies of shape and distribution effects for aligned continuous fibres, into the elasto-plastic deformation range. For axial straining the numerically predicted flow curve is found to lie about 7% above the simple R O M estimate 83
30,31
g
M
86
30
o =
fE t
F c
obtained by neglecting matrix hardening and the small hydrostatic stresses associated with the Poisson effect. Thus, the numerical results show no effect of fibre shape or distribution when fibres and matrix are strained in parallel. A n experimentally measured tensile curve for A l with 140 um diameter B fibres was in good agreement with the numerical calculation. This result is consistent with Isaacs and Mortensen's careful experimental study o f the axial tensile curves for A l with parallel continuous A 1 0 fibres, which, in contrast to Cu-W (Fig. 5), reveals no positive deviation from R O M estimates even for fibre spacings as small as 15 um. The fibre distribution effects on the transverse plastic response of the continuousfibre composite exceed those for the elastic response substantially. Thus the computations 18
2
3
232
30
Figure 17. A variety of numerical estimates of the transverse tensile curve for Al with 0.46% vol. pet. parallel continuous B fibres perfectly bonded to the Al matrix. Experimental results for Al-B with different interfacial conditions are added for comparison. 84
30
by Brockenbrough et al. show that the use of periodic square edge- or diagonal-packed fibre arrays does not provide estimates typical of the transverse tensile curve for composites with non-periodic phase structures. This problem can be overcome for continuous fibres, for which it is computationally feasible to handle unit cells with a fairly large number of fibres. The dotted curve in Fig. 17 labelled "random-packing" was computed using a unit cell, shown in Fig. 14d, containing 30 randomly positioned aligned fibres. For comparison we have added the transverse tensile curves for A l with 100 urn diameter B fibres measured by Kyono, Hall and Taya . It should be noted that metallurgically the curve is affected by the condition of the A l - B interface. Thus Kyono et al. found that the transverse curve is sensitive to heat treatment. A relatively mild heat treatment appeared to improve interfacial bonding (squares), while a more severe heat treatment led to deterioration of the B fibres and formation of a britde A 1 0 interface (triangles). Effects of debonding and other damage mechanisms have been included in analytical models as well as in cell model analyses . The results illustrate the importance of studying damage in terms of models with realistically specified conditions of matrix yield and interfacial decohesion as well as with unit cells large enough to simulate the considerable effects of shape and distribution of inclusions. 84
2
84
81
3
233 5. Summary and Conclusions A n attempt was made to discuss current modelling of the deformation behaviour of composite materials in the simplest possible terms without neglecting the experimentally revealed complexity of structure-property relations. The discussion covers thermoelastic and plastic behaviour, but modelling of the mechanisms of evolution of microstructures and damage was not considered. The following points were made: 1) The "rule of mixtures" (ROM) is capable of being extended to provide a simple and rather accurate model of the cyclic plasticity of aligned continuous-fibre metal matrix composites. Microstructural scale effects are revealed experimentally and they may be dealt with in terms of dislocation models of in-situ hardening and stress relaxation. 2) Effective medium theory provides a rigorous theoretical standard against which the mathematical soundness of structure-property relations predicted by composite models may be assessed in sufficiently simple cases. Such cases typically involve arbitrary phase structures possessing the overall symmetries of random arrays of particulates or of aligned fibres. 3) The extended R O M may thus be further extended into a "variable constraint model" ( V C M ) for composites represented by arrays of ellipsoidal inclusions with identical S tensors. The V C M reproduces the various bounds and self-consistent estimates delivered by effective medium theory. In addition the V C M delivers models of small strain plasticity typical of service conditions for composites. The models may be combined with dislocation mechanisms. 4) Complex structure-property relations were defined as relations not bounded by existing effective medium theory. Such relations are also provided by the V C M , but in general they involve arrays of inclusions with non-identical S tensors. Algebraic computer methods enable structural characterization of the distributions of inclusion shapes and orientations to be utilized in analytical modelling. It is of considerable interest to explore these novel possibilities of analytical modelling. 5) Complex structure-property relations also arise in numerical computer modelling where computational unit cells typically are greatly simplified compared with the "representative regions" of effective medium theory. Nevertheless, recent numerical cell models for the elastic deformation behaviour were found to be consistent with effective medium theory, except where deviations occur as a result of the use of periodic inclusion arrangements. In modelling of large strain plasticity of composites the cell models emphasize the complexity of plastic interaction of many inclusions via nonuniform plastic flow patterns.
234 6. Acknowledgements This overview was written as a part of the research carried out within the Engineering Science Centre for Structural Characterization and Modelling of Materials. One of us (BJ) is grateful to the Icelandic Council of Science (Vfsindarafl) for financial support. Thanks are due to Professor S. Suresh and his co-authors for access to reference prior to its publication. Further thanks are due to Professor A. Mortensen and Mr. P. Bystricky for stimulating discussions of parallel straining. We also thank Dr. S.L. Ogin for helpful discussions and Dr. S. Gudmundsson for assistance with algebraic computer codes. Figures 8, 9, 14, 16 and 17 were reproduced with the kind permission of the respective authors. 31
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14. 15. 16. 17. 18. 19.
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237 79.
80. 81. 82. 83. 84. 85. 86. 87. 88.
D. Juul Jensen, H . Lilholt and P.J. Withers, in Proceedings of the 9th Ris0 International Symposium on Metallurgy and Materials Science, eds. S.I. Anderson, H . Lilholt and O.B. Pedersen (Ris0 National Laboratory, Roskilde, 1988) p. 413. A . Kelly and G. Parkhouse, in Proceedings of ECCM-6, eds. A.R. Bunsell, A. Kelly and A. Massiah (Woodhead Publishing, 1993) p. 231. V . Tvergaard, in Proceedings of the 12th Ris0 International Symposium on Materials Science, eds. N . Hansen et al. (Ris0 National Laboratory, 1991) p. 173. N.J. S0rensen, A planar model study of creep in metal matrix composites with misaligned short fibres, to appear in Acta Metall. Mater. H . L . Cox, Br. J. Appl. Phys. 3 (1952) 72. T. Kyono, I.W. Hall and M . Taya, ASTM STP 964 (American Society for Testing and Materials, 1988) 409. A B A C U S Users Manual, Version 4.7, Hibbit, Karlson and S0rensen, Providence (1988). R. James, R. Lipton and A Lutoborski, S I A M J. Appl. Math. 50 (1990) 683. C.-H. Hsueh and P.F. Becher, J. Am. Ceram. Soc. 71 (1988) C-458. Z. Hashin and B.W. Rosen, J. Appl. Mech. 31 (1964) 223.
239 R A N D O M S T R U C T U R E M O D E L S F O R C O M P O S I T E M E D I A A N D F R A C T U R E
STATISTICS
D. J E U L I N Centre de Giostatistique, Ecole des Mines de Paris, 35 rue St-Honore 77305 Fontainebleau, France
ABSTRACT
As a consequence of microstructural heterogeneities, fluctuations in the mechanical properties of materials are observed in experiments. This requires a probabilistic approach to relate the microstructure to the overall properties, and to predict scale effects in the fluctuations of properties. In this paper, the problem of the strength of materials is addressed, and various models of random structures developed for fracture statistics are introduced. As any fracture criterion is sensitive to microstructural heterogeneities, such as flaws with low strength, or defects including a local stress concentration, large effects of small scale heterogeneities are observed for fracture phenomena. The approach combines the selection of appropriate fracture criteria and random structure models. It enables us to predict the probability of fracture of heterogeneous media under various loading conditions.
1. I n t r o d u c t i o n To predict the overall behavior of heterogeneous media from their structure is of importance i n many fields of applied sciences: i n materials science, this is the way to design and produce materials with customized microstructures, as far as the final use properties are concerned. W h e n considering the physical properties of heterogeneous media and their fluctuations at various scales, it is necessary to introduce appropriate techniques and models. T h i s problem is encountered in many situations, such as flows i n porous media, elastic properties, strength of composite materials... I t is solved by different methods (closed-form calculations, simulations), using i n any case models of random media. - T h e macroscopic (or effective) properties (like overall elasticity moduli of elastic composites) can be estimated from partial knowledge on the microstructure by variational methods. I n that case, approximations of average
240 properties of infinite random media are obtained, using homogenization techniques ' . — Another property of interest i n materials science is the strength, much more sensitive on local heterogeneities than the effective properties. F o r practical applications, it is of interest to calculate the probability of fracture as a function of the loading conditions and of the microstructure (Figure 1 ) . I t has many implications on the reliability analysis of parts and components at various scales, in many industrial fields (aeronautics, nuclear plants, civil engineering, etc.). T h e statistical distribution of the properties of finite domains c a n be obtained theoretically, as illustrated later. Scale effects, like the change of mean properties and of their variance with the size of specimens, can be predicted and compared to experimental data. 1
2
C a l c u l a t i o n of F i e l d a(x) P ( N o n fracture) = P
Microstructure Shape t h e part
Fracture strength
Loading Local fracture stress Sample
Change of scale
Figure 1 : Principle of Fracture Statistics calculations.
I n this chapter, some recent models for fracture statistics are reviewed. They enable us to predict fracture probabilities of materials under various loadings and on different scales. T h e approach is the following: - Choice of local (i.e. punctual) and of macroscopic fracture criteria; the first type accounts for crack initiation, and the second for the fracture of a specimen. — Construction of random structure models, defined on a point scale, for which the calculation of the fracture probability and of scaling laws is possible; this usually involves simplifications, such as the use of the stress field seen by an equivalent homogeneous medium, using the so—called local approach, as introduced by A. P i n e a u T h i s results into appropriate change of supports, as introduced below. 3
After a presentation of the used fracture criteria, each of them and proposed random function models with a point support are separately examined to cover
241 the following cases: fracture statistics of brittle materials; models with a damage threshold; crack arrest i n random media. 2. C h o i c e o f a f r a c t u r e c r i t e r i o n T h e first step required to develop fracture statistics models is the choice of l o c a l a n d o f g l o b a l f r a c t u r e c r i t e r i a . A local criterion is sensitive to the fracture initiation, while a global or macroscopic criterion accounts for the fracture of a specimen. 2.1. Local fracture
criteria
Various local criteria can be used: the fracture is initiated at points i n the structure where some intrinsic mechanical property of the material is exceeded, as the result of the applied load. Usually, this property is the c r i t i c a l s t r e s s a (x), or more generally the c r i t i c a l s t r e s s i n t e n s i t y f a c t o r K ( x ) for the tensile fracture i n linear elastic brittle m a t e r i a l s . When there is c o m p e t i t i o n b e t w e e n s e v e r a l f r a c t u r e m e c h a n i s m s , as for cleavage and intergranular fracture i n rocks and i n metals, multivariate criteria and multivariate random function models can be u s e d . T h e l o c a l f r a c t u r e e n e r g y v(x) corresponding to the creation of a fracture surface is used for crack propagations . c
I c
4,5,6
5 , 7
8,9
2.2. Global fracture
criteria
T h e following macroscopic fracture criteria, involving different fracture assumptions, were proposed by D. J e u h n ' . - T h e w e a k e s t l i n k m o d e l is well suited for the brittle fracture of materials, as in the cleavage of steel at low temperature ; it corresponds to a sudden propagation of a crack after its initiation. 4 , 5
7
3
- Models with a d a m a g e t h r e s h o l d generalize the previous one; they are valid for a fracture with several potential sites for crack initiation. - Models w i t h a G r i f f i t h c r a c k a r r e s t criterion compare, for each step of a crack path, the local fracture energy v(x) to the stored energy G(x) due to the deformation of the material. Formally, w h e n considering the overall condition of fracture of a sample, the first type of criterion uses a c h a n g e of s u p p o r t of the information by the operator A (infimum) over the domain of interest; the second family is connected to a change of support by convolution; finally, the last criterion involves a change of support by the operator V (supremum) over the crack front. I n the present approach, a n e q u i v a l e n t h o m o g e n e o u s m e d i u m with a random critical stress is used. T h i s simplification, which separates the applied
242
field a n d the c r i t i c a l field, enables us to o b t a i n closed f o r m results w i t h o u t any s i m u l a t i o n . T h i s is j u s t i f i e d for media w i t h a single component, l i k e polycrystals i n metals or i n rocks. However, t h i s approach cannot account for s m a l l scale stress f l u c t u a t i o n s induced b y the m i c r o s t r u c t u r e w h e n t h e components have different mechanical behaviors, r e s u l t i n g i n a s t r o n g c o u p l i n g between the applied stress field a n d t h e local fracture c r i t e r i o n . T h e t h r e e above—mentioned macroscopic fracture c r i t e r i a and t h e associated models are developed i n the f o l l o w i n g p a r t s o f t h e chapter. 3. F r a c t u r e s t a t i s t i c s m o d e l s f o r b r i t t l e m a t e r i a l s
3.1.
Introduction
Based o n t h e weakest l i n k assumption, these models assume t h e fracture of a p a r t , as soon as for a single p o i n t XQ , we have o ( x ) > o ( x ) ( F i g u r e 2 ) . I t corresponds to t h e i m m e d i a t e p r o p a g a t i o n o f a crack after i t s n u c l e a t i o n . To estimate t h e p r o b a b i l i t y o f fracture of a specimen B w i t h t h i s assumption, i t is necessary t o k n o w the p r o b a b i l i t y d i s t r i b u t i o n o f the m i n i m u m o f t h e values a ( x ) — a(x) over t h e loaded d o m a i n B . T h i s can be expressed as: 0
c
0
c
P no failure of B T h e stress field a(x) depends o n the local v a r i a t i o n s o f t h e c o n s t i t u t i v e law of the m a t e r i a l (namely o n t h e local elastic m o d u l i for a n elastic m a t e r i a l ) ; i t is therefore somewhat correlated to the local c r i t i c a l field o ( x ) . T h e calculation of the p r o b a b i l i t y P i n E q . (1) requires t h e f o l l o w i n g steps: c
- E s t i m a t i o n o f the local stress field, possibly u s i n g t h e f o r m a l expression based on t h e use o f Green's f u n c t i o n s ' ' 1 1
1 2
1 3
- B y t h r e s h o l d i n g the difference a ( x ) — a(x) to t h e value zero, one obtains a r a n d o m set A . One t h e n has t o estimate t h e p r o b a b i l i t y for t h e set B to miss the r a n d o m set A . T h i s is a p a r t of t h e general c h a r a c t e r i z a t i o n of t h e r a n d o m sets, as developed by G. M a t h e r o n ' . c
1
1 4
To t h e present a u t h o r ' s knowledge, t h i s p r o g r a m was n o t y e t f u l f i l l e d . I t is a tremendous task. T h i s can be accessed presently f r o m n u m e r i c a l s i m u l a t i o n s on realizations o f r a n d o m media as s h o w n for u n i d i r e c t i o n a l f i b e r composites . Otherwise, simplifications m u s t be i n t r o d u c e d . T h i s can be done for some r a n d o m s t r u c t u r e models w h e n u s i n g the d e t e r m i n i s t i c field a(x) seen by an equivalent homogeneous m e d i u m as i n the local a p p r o a c h . We use this s i m p l i f i c a t i o n i n the r e m a i n i n g p a r t o f t h i s chapter. 15
3
243
inf. {a (x ) - a(x )} c
0
> 0
0
Figure 2 : Illustration of the weakest link assumption; a) non fracture condition; b) fracture at point xo (from ref. 4). For a stationary random function a (x), the probability of no fracture of a specimen B under the deterministic stress field a(x) is given by: c
P { n o fracture of B } = P(a) = P{x E H ( o ) ] 0e
(2)
I n E q . (2), H (o~) is the set of implantations of the specimen B for which the minimum of the values a ( x ) — o(x) remains positive. I n general, the probability law (2) is not available. However it can be calculated i n a closed form for specific models of microstructures, as developed by D. J e u l i n ' ' ' ' . 0c
c
4
3.2. Examples
of stress fields
used for the weakest link
5
6
7
2 7
model
To illustrate the potential applications of the models, two kinds of stress fields are used i n E q . (1) and E q . (2):
244 — A u n i f o r m s t r e s s f i e l d applied on a cubic sample C with side L ( a ( x ) = a for c
x <= C ) . 0
— A n o n — u n i f o r m s t r e s s f i e l d i n a c r a c k e d m e t a l l i c s p e c i m e n (Figure 3 a), i n a n elastic—plastic material under plane—stress conditions
( a uniform
stress field a orthogonal to the crack plane is applied at the infinity). T h e crack tip is surrounded by a plastic region with radius 2Q (9). Outside the plastic region, the stress field o(r) (r = 0 at the point 0 located at the distance Q(0) from the crack tip (Figure 3 a)) is the elastic stress field for r > Q : at the point P (OP = r and (0P,6x)=8), we have
<
0i (r) =
K
(8)
w l 8 y
with the stress intensity factor K j for a large size plate with a central crack with length 2a orthogonal to the tensile stress : K
T
=
ojna
For a given crack size a, and the macroscopic stress o, the region of the specimen (e.g. a cube of side L ) under plastic yielding is a cylinder with axis L and with section ,2
B (B =B(l)beingobtainedfor
a = 1; we will neglect the changes of Q with the V
angle 9).
y
/
F r o m now, we make the two following assumptions : i) fracture by cleavage can only be initiated into the plastic region B , when a local critical cleavage stress a is reached. c
ii) inside the plastic region B, the variable stress field (Figure 3 a) is replaced by an homogeneous stress field equal to the maximum stress hj
y
(Figure 3 b). With this
simplification, the probability of fracture for a given applied stress a is overestimated. B u t the general line of the results will not be altered. It is instructive to compare the behaviour of a sample under a uniform stress field and of a cracked specimen: i n both cases for our approach, the domain where fracture occurs is under an homogeneous stress field. I n the first case the part of the material where o>o' is the sample itself. I n the second case, the region of interest is the plastic region submitted to the homogeneous field Xa (independant y
on the applied stress a, and constant for a material with a constant yield stress 0 ) . y
T h e size of the region increases with the applied stress a and with the crack length 4
2
a, and its volume is proportionnal to (<j-) a . So the macroscopic stress a induces a 4
volume effect proportional to o .
245
Figure 3 : Stress field at the crack tip of an elastic-plastic material; a) plane stress conditions; b) simplified stress field (constant in the plastic zone) for the calculations, (from ref. 4 and 27) Therefore, it is expected, that each kind of applied stress field will bring its own piece of information on the random critical stress field o (x) i n the material. T h i s will be illustrated i n the next sections for different models o (x). c
c
246 I n addition any change of the yield stress a due to specific conditions (e.g. a decrease with temperature) can be accounted for in our results, and will modify fracture stratistics of cracked specimens. y
3.3. The Boolean random
varieties and the weakest link
model
T h i s section presents a wide class of models for which fracture statistics corresponding to the weakest link model are available. T h e i r field of application is not restricted to micromechanical problems. Other models, derived from the D e a d L e a v e s random functions, are proposed by D. J e u l i n ' ' and are not presented here. 4
3.3.1 Construction
of the Boolean random
7
2 7
varieties
T h e B o o l e a n r a n d o m v a r i e t i e s describe structures with different geometrical defects: points or grains, fibers, strata. T h e y are generalizations of the w e l l - k n o w n Boolean model proposed by G . M a t h e r o n as a model of random set, initially used for simulating porous media. I n the present application, they are used as random models to represent the critical stress field a (x). I n fact, as for other models introduced later on, it is possible to use them (or their multivariate version) to simulate any physical properties of composite media, such as for instance their moduli of elasticity. 1 , 1 4
c
For these models, it is possible to derive theoretical models of change of scale including size and shape effects (concerning the specimen B ) , and also microstructure effects. We will recall here the general results and illustrate them for the uniform stress field and for the stress field developed at the tip of a macroscopic crack in elastic—plastic materials. T h e Boolean random Varieties can be constructed as follows (Figure 4 ) : • i n R , n Poisson linear Varieties, with dimension k (k = 0,..., n - 1 ) can be defined . A geometrical interpretation of the varieties is the following : consider a Poisson point process ^(co) with intensity 0 (dca)(j. _ (dx) on the linear varieties of dimension (n—k), with the orientation co, and containing the origin ( | i _ is the Lebesgue measure i n IR ) . I n each point X;(co) of the process, a variety of dimension k, V (co) ., orthogonal to direction co, is located. A realization of the variety of dimension k is obtained as n
14
k
n _ k
n
k
k
x
v = I J V (to) k
k
X,(<0)
Xi
n
k
1) on Poisson points
-x
x
x
x
X
xk 2) on Poisson planes
jtk(co ) 0
x k : Poisson points
0(co)dco on D(o))
Boolean strata 3) on Poisson lines
0
3
Figure 4 : Construction of Boolean random fields in R . (from ref. 4)
248
3
For f r a c t u r e statistics, we operate i n E , so t h a t w e consider t h e f o l l o w i n g Poisson varieties : points (k = 0), lines ( k = 1), a n d flats ( k = 2 ) . I n a d d i t i o n , we w i l l U m i t ourselves to isotropic models, b u t anisotropic s t r u c t u r e s ( w i t h appropriate 0 (duj)) can be used w h e n necessary. k
• i n a second step, we consider independent p r i m a r y r a n d o m functions Z'(x), lower semi—continuous, w i t h a subgraph a d m i t t i n g compact sections almost surely a n d w i t h Z ' ( x ) < a . S t a r t i n g w i t h a homogeneous m e d i u m w i t h s t r e n g t h Z ( x ) = a , we b u i l d t h e Boolean r a n d o m V a r i e t y o f d i m e n s i o n k as follows : m
m
Z(x) = A { Z ' ( x -
y i
) ;y, G
;X; G 9> }
(3)
k
Figure 5 : Simulation of a Boolean random function in the plane (512x512x8 bits), using the supremum of conical primary random functions with radius 45 pixels (from ref. 10).
W i t h t h i s construction, various types o f defects ( w i t h a l o w e r etc) i n a homogeneous m a t r i x are obtained: r a n d o m g r a i n defects (negative o f F i g u r e 5 i n the plane), fiber defects (negative o f F i g u r e 6 i n the plane), a n d s t r a t a defects ( i n d u c i n g fibers b y plane sections). For these models, t h e p r o b a b i l i t y o f fracture of E q . (2) is expressed as: P{no failure of B } = exp - 0 p ( H | , ( o -
a )) m
(4)
249
Figure 6 : Simulation of Boolean random fibers in the plane (512x512x8 bits), using the supremum of conical primary random functions with radius 15 to 25 pixels (from ref. 10).
I n E q . (4), u. is equal to volume V (for grains), (IT/4)S (S being the surface of the area) for fibers, or to the integral of mean curvature M for strata (with 1 f 1 1 M — I ( = - + -~-)dS, R i and R2 being the main radii of curvature over the =
surface area S of the set H ^ o — o ) . It is assumed in E q . (4) that H ^ , (a — a ) i s a convex set for the case of fiber and strata defects. m
r a
I n E q . (4), the average measure j l is taken over the realizations of Z'. A slightly more general formulation of E q . (4) can be obtained, starting from a family of primary random functions Z ' , implanted on random Poisson varieties Vk(t) with intensity 6(t) (t being equivalent to a time parameter): t
P{no failure of B } = exp -
(1(11^(0- - a ))6(dt) ra
(5)
J R
T h e general properties of these models for fracture statistics applications are the following: • B y construction, the fracture stress allows correlations at a microscale (depending on the size of the support of the defects Z't(x)).
250 • W h e n considering the fracture strength at a macroscale (namely the scale of the specimen B ) , the following effects are expected to be observed:
size effect
(through the set of points H^. (o — o ) , depending both on B and on Z ' t ; shape m
effect for the specimen B ; microstructure effect (grains, fibres or strata give different weights to the probability of failure, through y S or M ) . 3.3.2. Some examples of Boolean
Varieties
We develop now some specific examples of models, useful for applications, based on two separate constructions, and with o = + oo. Theoretical expressions for a homogeneous stress field are given. — F i r s t we consider defects Z ' with a random closed set support X'n. T h e critical stress of the defect a is a random variable Z', independent of X'n, implanted in space with the intensity 6 ( 0 ) . T h e resulting field o (x) is a mosaic made of domains where a is constant. It is a good simulation of a multiphase material containing grains with different sizes and strengths. Since we are interested in brittle fracture i n traction, we restrict ourselves to the scalar case with a > 0 (a being here the local maximal principal stress). I n the case of a uniform stress field a over the specimen B, the distribution of the fracture strength OR of B can be deduced from E q . (5) as : m
t
c
c
c
P{o
R
> a] = P{no failure of B } = exp^ - TJ ( X ' © B)<J)(o) j
(6)
0
I n E q . (6), TI is the average over the realizations of the random closed set X ' n X'n © $ = (x; B
x
;
(~) X ' * 0} is the result of the dilation of the set X b y the set 0
0
1 6
B , and
rc
0(t) dt (7) h I n practice, any increasing non-negative function (b(o) can be used i n E q . (6). We illustrate this with well—known examples. For defects with intensity 6 ( 0 ) = m8(a - a )
m _ 1
0
m
get
P{a
R
R
with m > 1 and a > o , we 0
obeys a Weibull distribution, with aj? = 1/0 :
> a} = exp - p ( X ' © B ) ^ - ^ 0
(8)
T h i s distribution is very popular amongst practitioners of fracture statistics, as can be seen from the numerous r e f e r e n c e s ' . T h e classical version of the distribution is limited to point defects X'n, where the more severe defects (with a low o ) are much sparser than the less critical defects. T h i s model is expected to be 17
18
c
suited to high grade materials with a thorough control of defects.
251 F r o m E q . (8), it is easy to calculate the expectation E [ a ] and the variance of a , as a function of V = p ( X ' © B ) : R
R
0
E[OR] = o + V - V - W X l + 1/m)
(9)
0
D [o ] = o (V) 2
2
2
R
ra
/ (r(l
2
+ 2/m) - T (1 + 1/m))
x
(10)
1
where T(x) is the E u l e r i a n function T(x) = 1 t exp — t dt. Jo When oo = a, the coefficient of variation of o , D [ a ] / E [ a ] does not depend on y which can be easily checked from experimental data : R
R
R
D i n 1/Efcr 1 - ( H I + 2/m) - T ( l + l / m ) ) V D[o ]/E[a ] ( l 4- 1/m) 2
R
R
2
F
E q s . (9) and (10) express a scale effect for the expectation and the variance of the strength, which always decrease with the size of the specimen, but differently according to the kind of Boolean Variety. T h i s scale effect is also observed on the median a of the fracture strength : M
o
= a
M
1
+ K V- /™
0
(12)
Similarly, we c a n use a bimodal Weibull distribution, where there is a superposition of defects from two Weibull populations (with parameters (0i, oni, m i ) , and (02, 002, 0 1 2 ) ) ; i n that case, cb(o)
= 0,(0- -
o
• For defects with intensity 0(a) =
0 1
)
m l
+ e (o
-
2
M
0
O 2
) '
(13)
the distribution of strength is exponential,
and can be considered as a particular Weibull distribution, with m = 1. P{o
R
> a] = exp - ( 0 V o / a )
(14)
u
2
with expectation
/a \ variance l ^ j and coefficient of variation 1.
• For defects with intensity
0(a) = ^
9
m
_ i) m a
cp(o) = —3— ^—r when m > 1. ^ I m-l m-l • For defects with intensity 0(a) = 0 / a for a > a a
f
o
r
a
- °o
>
°>
w
e
h
a
v
e
a
p{a
R
> a) = ( a / a ) 0
0
> 0, we get (15)
252 for a > a and V = TX(X' ffi B ) , as previously. 0
0
Contrary to the Weibull model, the last two populations of defects are dominated by defects with very low critical stress o . c
T h e distribution given i n E q . (15) is well known as the distribution of P a r e t o
19
Its expectation and variance are infinite for 0 V < 1 and 0 V < 2 respectively, namely for specimens B with a low size. W h e n 0 V > 1, we have
For 0 V > 2, D
K 1
= (0V-2K0V-1)
(
1
7
)
{
1
8
)
and D
e
K ) / ( ° R ) =
yevfflv"-2)
T h e last properties, given by E q s . (16—18), decrease with the size of the specimen B . T h e size effect is higher for Pareto's model t h a n for Weibull's model. • When a>(o") is a sigmoidal function, a saturation effect is observed on the cumulative distribution of defects :
0
(19)
exp
for o > Or S u c h a distribution was observed for the fracture of carbon f i b e r s
20
: using a
multifragmentation test on a single carbon fiber embedded i n a epoxy matrix (Figure 7 ), it is possible by means of accoustic emission to estimate
the
cumulative distribution of the critical stress of defects on fibers. Figure 8 shows a micrograph of the fiber after reaching the saturation limit (each fracture being followed by a decohesion between the matrix and the fiber over a given length, a limited number of defects can be sampled until this saturation limit). O n Figure 9 are shown the cumulative functions o)(o) observed on two types of fibers. As opposed to conventionnal determinations of defects statistics by means of single fracture tests on fibers (for which the weakest link assumption can be applied!), a much larger sample of defects is accessed; for larger critical stresses, it departs from the power law function a>(a) involved i n the Weibull model, which is recovered for the lower stresses by a Taylor's expansion of E q . (19). T h e results obtained in Figure 9 are introduced i n finite element simulations to study the 15
damage of fiber—matrix composite l a y e r s .
T
T
T
~i i i
1
1 Single Fibre Specimen
First Breaks
Saturation Limit
Figure 8 : Multifragmentation of a fiber, after reaching the saturation limit (from ref 20)
254 600
0
1000
2000
3000 4000 5000 Fibre Stress (MPa)
6000
7000
8000
T300 Carbon Fibre, comparison between theoretical and experimental cumulative density function, (single fibre specimen)
600
0.00
2000.00
4000.00
6000.00
8000.00
10000.00 12000.00
Fibre Stress (MPa)
M40 Carbon Fibre, comparison between theoretical and experimental cumulative density function (single fibre specimen) Figure 9 : Estimation of the cumulative intensity of defects (sigmoidal function given by E q 19) from the data obtained on carbon fibers by the multifragmentation test (from ref 20).
255 - A second f a m i l y o f Boolean Varieties models uses defects made o f r a n d o m closed sets w i t h a constant c r i t i c a l stress a as previously, b u t the value o f a depends o n t h e r e a l i z a t i o n o f t h e p r i m a r y g r a i n X ' n , as follows. We consider r a n d o m g r a i n s X ' ( d ) (homothetics w i t h r a t i o d o f r a n d o m grains X ' = X ' ( l ) ) . For each g r a i n , o = tp(d). I n fracture mechanics, i t is c o m m o n t o assume t h a t ijj(d) = K / v/d, each g r a i n X ' ( d ) b e h a v i n g l i k e a crack w i t h l e n g t h d. For the model, w i t h X ' ( u ) = u X \ we have, i n t h e case o f a homogeneous stress field o : c
c
c
P{a
R
> a) = exp -
I[
66(u) (u)u(X'(u) © B)du
'KVo
I n E q . (20), w e can use Steiner's f o r m u l a convex: for Boolean grains
1 4 , 1 6
w h e n the sets B and X ' ( u ) are
2
V(X'(u) © B) = V(B) +
+ ^jpU S + Vu
3
(21)
for Boolean fibers S ( X ' ( u ) © B ) = S(B) + ^
u
M
2
+ uS
(22)
for Boolean s t r a t a M ( X ' ( u ) © B) = M ( B ) + u M
(23)
I n Eqs. (21) —(23), V, S, a n d M are t h e average v o l u m e , surface area, and i n t e g r a l o f m e a n c u r v a t u r e , over t h e realizations o f t h e r a n d o m set X ' . D e p e n d i n g o n t h e choice o f 0(u) (i.e. o f t h e g r a i n size d i s t r i b u t i o n of the model), various d i s t r i b u t i o n s can be obtained. W h e n u.(B) - » °°, we recover asymptotically W e i b u l l ' s d i s t r i b u t i o n for 0(u) = 0 u ( w i t h m > 4 for grains, m > 3 for fibers, a n d m > 2 for strata), a n d Pareto's d i s t r i b u t i o n for 0(u) = 0 / u . _ r a
T h i s t y p e o f m o d e l was used for t h e f r a c t u r e o f a silicon n i t r i d e ceramic b y C. Berdin . I n t h i s m a t e r i a l i t was s h o w n f r o m fractographic analysis t h a t the b r i t t l e f r a c t u r e occurs on t h e most severe defects (inclusions a n d porosities w i t h a size r a n g i n g b e t w e e n 20 a n d 70 u m , as i l l u s t r a t e d i n F i g u r e 1 0 ) . T h e size d i s t r i b u t i o n o f defects was estimated f r o m image analysis o n polished sections, using a n o p t i c a l confocal microscope. U s i n g a spherical shape assumption, i t was possible t o estimate t h e d i s t r i b u t i o n o f diameters b y means o f a standard stereological r e c o n s t r u c t i o n p r o c e d u r e . T h e e x p e r i m e n t a l d i s t r i b u t i o n s were fit to t h e f o l l o w i n g t h r e e - p a r a m e t e r s model, w i t h a power l a w t a i l for t h e most severe defects: 2 1 - 2 3
24
256
T h e assumption of a Poisson distribution of defects was validated from the study of the statistics of the number of defects per field of m e a s u r e m e n t . A procedure of identification of the model was proposed. I t combines image analysis data and the strength data obtained for various mechanical tests (4 points bending, biaxial bending, and tensile tests), using the stress distribution induced from the defects size distribution given by E q . (24). F o r simplification, the size of the defects is neglected, as compared to the size of specimens, so that they are replaced by points. A s illustrated by Figure 1 1 , a good fit could be obtained, when accounting for the differences of microstructure observed between the tensile test specimens and the others. 23
<
> 100 um
Figure 10 : Micrograph of a silicon nitride ceramic showing porosities and inclusions, where brittle fracture is initiated (from ref 22).
257 1j
density (/mm3) 100000
Pr
0,8 -• 10000 0,6 -•
1000
4¬
0,4
100
0,2
10
a (MPa)
H
1 0
5
1
1
1
1
1
1
1
1
10 15 20 25 30 35 40 45 diameter (microns. Microstructure. Figure
5
10
•+-
-f-
300
400
500
600
700
800
900
Fracture probability. : Specimens for 4 Points bending.
density (/mm3)
0
0
1 T Pr
15 20 .25 30 35 40 ,45 diameter (microns)
Microstructure. Figure
300
400
500
600
700
800
900
Fracture probability : Specimens lor biaxial bending. 1 -r Pr
density (/mm3) 100000 10000 1000 100 10
H
0
5
1
1
1
1
1
1
1
1
10 15 20 25 30 35 40 45 diameter (microns)
300
400
500
600
700
800
900
Microstructure. Fracture probability. Figure : Specimens for tensile tests. Figure 11 : Comparison between the experimental data and the models obtained for the probability distribution of the diameters of defects, and for the cumulative probability distribution of strength; i) 4 points bending; ii) biaxial bending; m) tensile tests (from ref 22, 23).
258 3.3.3. Fracture statistics of a cracked elastic—plastic
specimen
To get m o r e simple notations i n t h i s p a r t , we consider Boolean varieties (for i ) grains, i i ) s t r a t a a n d i i i ) fibers) u s i n g a p r i m a r y r a n d o m f u n c t i o n Z ' . We note X ' t h e set o f p o i n t s x where Z'(x) S z. We assume t h a t t h e sections X ' are convex. 2
z
We can apply E q . (3), w i t h t h e stress a = Xy < o , a n d w i t h t h e d o m a i n B restricted to t h e c y l i n d e r generated b y t h e plastic r e g i o n defined i n section 3.2. For a cylinder w i t h section B and h e i g h t L , we have : m
I
V = L A(B) S = JL(B)L + 2 A ( B ) M = jt(L + i j . ( B ) ) 2
(25)
L ( B ) and A ( B ) being the perimeter and the area o f the plastic region (in section). F r o m E q . (3), we get (stating b = 1 - G ( a - hy ) and o - Xo - on. m
i)
exp
+
9
L
y
m
y
4a2bA(B)
! " [ [fe)
+
2
^ ( b ^ W M ^ o )
+ f (X'o ) 0
2
A f e ) a A(B)M((X'a ) + | ( ^ ) a i , ( B ) S ( X ' o ) + V(X'a ) 0
0
ii)
exp
- e(bjc(L + i f ^ J ai,(B)) + M ( X ' o ) )
iii)
exp
-
0
0
9|
a ) abJH(B)) +
M(X'a ) 0
y
+ 2 (^)VbA(B)4-(^)UB)M^£o)
+
s(X'a ) 0
(26) F r o m E q . (26), i t is clear t h a t a cracked specimen of a n e l a s t i c - p l a s t i c m a t e r i a l nearly follows a W e i b u l l d i s t r i b u t i o n w i t h t h e power 4, at least f o r large specimens. T h i s l i m i t s the influence o f the d i s t r i b u t i o n o f t h e defects c r i t i c a l stress o n t h e size effect, as seen now.
259 3.3.4. Size effects T h e study of size effects is very important from a practical point of view. I n the present context, it describes the change of statistical properties of the strength when increasing the size of samples. T h i s has many consequences i n relating data measured at the laboratory scale to the large scale behavior of parts or even of buildings... F r o m a n experimental point of view, it is not difficult to examine size effects by considering the change of the mean or of the median strength with the size of specimens. For the weakest link models, the size effect is the decrease of the median strength with the volume of the specimen, since the probability to observe a critical defect increases with the size. Its analytical shape depends generally on the choice of the model and on the statistical properties of the defects (size, shape, critical stress). We give now some results derived i n the case of the Boolean random v a r i e t i e s . 4,27
We will now compare the effect of a uniform stress field, or of a non—uniform stress field developped i n a cracked specimen on the size effects for the Boolean varieties. To give explicit results, we will consider particular primary functions, where Z'(x) admits a set of maxima which is a random variable Z' (Z' < o ) with probability law G ( z ) : m
G(z) = P{Max (Z'(x) ; x £ X'o] < z}
(27)
I f we use the same Z'(x) for the three kinds of Boolean varieties, with the distribution function G satisfying 1 - G(o
m
Q
- a) = ( ^ )
(a < a ) m
a > 0
(28)
the median value O M of the macroscopic convex samples under a uniform tension(or any quantile of this distribution) will change with their volume V in the following way : i) o
1
= K V" /"
M
ii) a ii) o
M
—K V-V** 2
M
3
= K V" / "
(29)
T h e median always decreases with the size of the sample; the size effect increases with the type of structure i n the following order : Boolean strata, Boolean fibers, Boolean functions. A s t r a t i f i e d m e d i u m s h o u l d b e l e s s
260 sensitive
to
size
effects
in
fracture, when
the
stress
field
is
h o m o g e n e o u s (and w h a t e v e r t h e h i s t o g r a m G(z) i n t h e m o d e l ) . The size effect of the cracked specimen can be studied f r o m the basic Eq. (26) as a function o f the length L . I t is clear that the probability o f fracture at the level a w i l l decrease with L (as the probability to encounter a weak zone w i l l increase). For large specimen, we have t h e f o l l o w i n g a s y m p t o t i c r e s u l t s o n t h e size effect (L-co) :
i) i f we can neglect t h e size o f t h e sections X '
0 o
o f t h e p r i m a r y f u n c t i o n Z'(x) as
compared to t h e plastic r e g i o n B , we get f r o m E q . (26i) : a
=
M
K L - * =
mm)
KV-TV
for a cube o f side L . i i ) For Boolean strata, i t is easier t o express t h e size effect as a change o f the average fracture stress a w i t h L . We have f r o m E q . (30 i i ) , for a n y m i c r o s t r u c t u r e : a — K e x p — c L = K e x p — cVs for a cube o f side L w i t h C = 0 n ( l — G ( o
m
(30ii)
— Xa )). y
i i i ) w i t h t h e same conditions as i n case i ) (we can neglect M(X'<, ) as compared to 1(B)), we have t h e f o l l o w i n g size effect for t h e Boolean fibers : 0
a
M
= K L 4 = K V - J
(30iii)
for a cube o f size L . T h e a p p r o x i m a t i o n s i n v o l v e d i n E q . (30i) a n d (30iii) are v a l i d for Boolean s t r u c t u r e w i t h zero range. E q . (30) shows t h a t t h e a s y m p t o t i c s i z e e f f e c t s do n o t d e p e n d o n t h e d e t a i l s o f t h e d i s t r i b u t i o n f u n c t i o n G ( o _ ) (as i t was the case for a u n i f o r m stress field i n t h e specimen). I n p a r t i c u l a r t h e size effect (30i) recovers t h e results o f B E R E M I N : large cracked elastic plastic materials follow a W e i b u l l d i s t r i b u t i o n w i t h exponent m = 4. m
a
2 5
T h e s e n s i t i v i t y to size effects (and n a m e l y t o the l e n g t h o f t h e crack front) depends o n t h e type of m i c r o s t r u c t u r e used to describe the d i s t r i b u t i o n o f defects; i t increases i n t h e f o l l o w i n g order : Boolean functions, Boolean fibers, Boolean strata. We therefore get a n opposite conclusion t o t h e homogeneous stress field case :a s t r a t i f i e d m e d i u m should be m o r e sensitive to t h e size effect i n f r a c t u r e for a cracked specimen. T h i s results f r o m a h i g h e r p r o b a b i l i t y for t h e crack f r o n t to hit a low strength domain. I t is possible to d r a w the f o l l o w i n g p r a c t i c a l conclusions f r o m t h e comparison between t h e u n i f o r m stress field and t h e cracked specimen, i n o r d e r to m i n i m i z e t h e size effect o n t h e a v e r a g e s t r e n g t h of t h e m a t e r i a l : — for parts w i t h o u t cracks, u n d e r a n homogeneous stress field, a stratified m e d i u m should be less sensitive to the size o f t h e p a r t ;
261
- when cracks may be present i n a specimen, a Boolean microstructure (i.e. a structure with finite range, where the domains with lowest strength have a finite size) should be less sensitive to the size of the part. - when one wants to minimize the size effect for the two situations (cracked or uncracked sample), a good compromise would be to use a material with a fibrous distribution of lower stength domains. 3.4. Competition
between fracture
mechanisms
I n practice, the fracture process of materials is often very complex, and may result from a competition or a cooperation between various mechanisms. It is therefore of importance to work out models based on a multi - criterion approach, as illustrated now. T h i s generalization offers no theoretical difficulty. I t only implies a n extension of the number of parameters involved i n the model, which requires more data for their estimation than i n the case of more simple models. 3.4.1. Generalization
of the weakest link model
T h e fracture of a material may involve a c o m p e t i t i o n b e t w e e n v a r i o u s m e c h a n i s m s (for instance between cleavage and intergranular fractures in steels ). T h i s competition may result from different stress criteria (such as tensile, versus shear), or from different populations of defects. I n the first instance, we must consider multivariate fields o'(x) and o (x) (i = 1,2,..., n for n mechanisms). I n that case, the probability law given i n E q . (2) becomes : 26
c
27
P[no failure of B } = y
= P
A
(oJ(x) - a\x
x£B,
- y)) > 0,...,
n
A ( a ( x ) - o"(x - y ) ) > 0
(31)
xEB,
Note that i n general the random functions involved i n E q . (31) are not independent. Appropriate multivariate random field models are available for such situations . When different populations of defects are present i n the material (corresponding either to the same fracture stress criterion, or to different fracture stress criteria as i n part 4 below), we may encounter the following situations : (i) the spatial distribution of defects is so heterogeneous that a single population is present i n each specimen (defect k with probability pk), (ii) the spatial distribution of defects is very homogeneous, two separate populations being either correlated (see section 3.4.2) or independent (the defect k occurring with probability pk), 7
262 (iii) the specimen V is made of disconnected subdomains V ; ; each V i contains a single class of defects i. F o r small volumes V, (iii) is equivalent to (i). I f we consider for example the weakest link assumption i n a homogeneous stress field o, distributed according to a Poisson point process (which is a particular Boolean random function), we have : T ( o ) = P{no failure of B k
y
by defects k} = e x p { - V ( B ) tt> (o)} k
(32)
For situation (i), we get: k=n
T(o) = P{no failure of B } = £ y
k=n
p T (o) = £ k
p e x p { - V(B)dj (o)} (33)
k
k
k
k=l
k= l k=n
For a small volume Y T ( a ) = 1 - V ( B )
X
Lk=i
Pk * k ( ° )
I n case (ii) and for
independent defects, we obtain : k=m k
n
T(o) = n T ( o ) = exp - V ( B ) k=l
k
X
p
0
k ^kC )
(34)
Jt-1
For small values ( V - * 0 ) , situations (i), (ii), (iii) become equivalent. I n case (iii), k=n
T(o) = exp - V ( B )
X
(35)
4> (o) k
For large volumes V, E q . (35) is equivalent to E q . (34) (situation (ii)), with Pk = V / V . k
Specific models of this type can be developed on this basis for applications, as seen now for polycristals. 3.4.2. Example mosaic
of application
to a polycrystal
modelled
by a Voronoi
random
As an illustration, we now consider an example of model introduced by D. J e u l i n for the case of polycrystal materials, such as steels or ceramics. We will present results obtained for a single fracture criterion, and for a competition between cleavage and intergranular fracture. Approximate results, valid for large specimens, are given. 7
— A mosaic model is obtained from a random tesselation of space. E a c h class of the tesselation may represent grains of a polycrystal, while its boundaries can model grain boundaries. For cleavage fracture, one may assume that the fracture
263 of any grain is obtained for a random critical stress o following a probability law c
F(o) = P { o < o } . F o r two grains, independent realizations of o are considered. c
c
I n this model, we recover the weakest link assumption for each crystal (after initiation i n a grain, a crack propagates and stops on the grain boundaries). We call G B ( S ) and G ( s ) the generating functions of the random number of grains contained i n the sample B , N ( B ) , and of the number of grains in B , N(rj), where o
c
< a (for simplification, we assume a homogeneous stress field o(x) = a i n a n = oo
stationary random tesselation) : G ( s ) = B
V
n
P(N(B) = n] s .
n= 0
G(s) is obtained i n two steps : when N ( B ) = n, N(o) is a binomial random variable with
parameter p = F ( o ) and generating
function
(ps + 1— p )
n
Therefore G(s) = Grj(ps + 1 - p ) , from which p can be calculated. n
— T h e probability po, required for the application of the w e a k e s t l i n k m o d e l is obtained by po = GB(1—p). For example, the V o r o n o i m o s a i c is built from the 3
zones of influence of a Poisson point process i n R approximately N ( B ) = N + l ,
where N follows
(with intensity 8). We have
a Poisson distribution with
parameter 8' = 8V(B). F o r this model: G ( s ) = sexp8'(s - 1)
(36)
G(s) = (ps + 1 - p)exp8'[p(s - 1)]
(37)
B
Therefore
PO = (1—p) exp(—8'p), and for the weakest link assumption, P{o
R
> o) = (1 - F ( o ) ) e x p ( - 8u (B)F(o))
(38)
n
The scaling laws i n E q . (38) depend on the distribution F ( a ) . For a m a c r o s c o p i c c r a c k i n an elastic plastic material (a — Xa i n the plastic zone), y
with m
0
=
F(ka ). y
P{o
R
a o) = (1 - m ) e x p j - 8 m 0
2
0
La A^j
j
(39)
We obtain for a specimen containing a crack with length 2a the W e i b u l l d i s t r i b u t i o n with parameter m = 4, independently on the distribution F ( o ) , as already obtained i n the case of the Boolean varieties. -
A c o m p e t i t i o n b e t w e e n t w o f r a c t u r e m e c h a n i s m s , namely cleavage and
intergranular fracture, can be easily introduced from the mosaic model. As before, we use a distribution F ( o ) for the fracture of grains by cleavage. I n g
addition, a random fracture stress with law Fj(o) is independently affected to each interface. T h e number of grains N
g
and of interfaces Nj i n the sample B are
264 correlated, 1 = N
g
since
- N; + N
e
they
satisfy
the
Euler
equation:
1R ,
- N s , w h e r e e a n d S are edges a n d s u m m i t s o f t h e tesselation.
I t is k n o w n t h a t for a tesselation i n e q u i h b r i u m i n TR interfacial
3
in
Nj/Ng = 7
energy)
2 8
3
(i.e. m i n i m i z i n g the
. For c a l c u l a t i o n , w e need
the bivariate
g e n e r a t i n g f u n c t i o n GR(SI,S2) o f N ; a n d N , t o estimate t h e d i s t r i b u t i o n o f the g
n u m b e r o f m i c r o s t r u c t u r a l elements (grains + interfaces), w i t h g e n e r a t i n g function G(s)) where o p
c
< a. I n t h e case o f a homogeneous stress field, w i t h
= F ( o ) a n d p i = F,(a), for N
g
g
a n d interfaces w h e r e o
c
= n
g
a n d N i = n i fixed, t h e n u m b e r s o f grains
g
< a follow b i n o m i a l d i s t r i b u t i o n s w i t h p a r a m e t e r s p
a n d pi, a n d w i t h g e n e r a t i n g functions (pgS + ( l - p ) )
n g
g
g
m
a n d (pi s + ( l - p ; ) ) . The
sum of these t w o variables admits as a g e n e r a t i n g f u n c t i o n t h e p r o d u c t o f t h e two, a n d therefore G ( s ) = G B ( p s + l - p , Pi s + l - p i ) . g
g
T h i s can be simpUfied i f N ; = a N ( a = 7 o n average i n TR ) : 3
g
G(s) = G [ ( p B
i S
+
1
-
P
I
) >
G
+
1
- p )]
(40)
g
I n E q . ( 4 0 ) , G B ( S ) is t h e g e n e r a t i n g f u n c t i o n o f t h e n u m b e r o f g r a i n s h i t b y the sample B . For a V o r o n o i mosaic, as before, G B ( S ) is a p p r o x i m a t e l y given b y Eq. and :
(36),
G ( s ) = (PJS + 1 - p ) (PgS + 1 a
;
P g
)exp9'[(p s + 1 i
) (pgs + 1 - p ) - l ] a
P i
g
(41)
T h e weakest l i n k m o d e l is obtained f r o m po = G ( 0 ) : P{a
> a) = ( 1 - F ^ n i - F (o))exp{e x (B)((l - F ^ H l - F ( a ) ) - 1 ) }
R
g
(
n
g
(42)
I n t h e case o f a macroscopic crack i n a n elastic—plastic m a t e r i a l , we use i i ( ^ y ) a n d m = F ( t a j ) i n E q . ( 4 2 ) . F r o m t h e v o l u m e of t h e plastic zone, we get again a W e i b u l l d i s t r i b u t i o n w i t h p a r a m e t e r 4 , as i n t h e case o f E q . ( 3 8 ) . m
=
0
F
CT
g o
g
y
For applications o f these models t o r e a l m a t e r i a l s , t h e f o l l o w i n g data are r e q u i r e d : G B ( S ) or G ( s i , s ) ( m i x e d f r a c t u r e ) , F i ( o ) , F ( a ) , a. I n simplified versions (the V o r o n o i model, or a constant n u m b e r o f grains a n d interfaces h i t by B, t h a t can be estimated b y image analysis a n d stereological techniques), the h i s t o g r a m s F , a n d F m a y be estimated f r o m e x p e r i m e n t s i n v o l v i n g a single f r a c t u r e m e c h a n i s m , a p p l y i n g Eq. ( 3 8 ) . T h e n t h e assumptions o f t h e m o d e l can be tested f r o m a comparison between e x p e r i m e n t a l f r a c t u r e statistics a n d the t h e o r e t i c a l predictions deduced f r o m E q . ( 4 2 ) . B
g
2
g
265 4. F r a c t u r e s t a t i s t i c s m o d e l s w i t h a d a m a g e
threshold
It is possible to generalize the weakest link criterion i n various ways, as shown below - - . 6
7
10
When relaxing the weakest link assumption, it is possible to propose critical damage models, based on a critical volume fraction, or on a critical intensity, depending on the nature of the defects. 4.1. Fracture
statistics
models with a critical
volume
fraction
We consider i n a bounded specimen B a stress field o(x) applied to a material with the critical stress field o (x). T h e indicator function l (x) of the domain D is defined a s : c
a<:
l (x) = 1 if a (x) < o(x), else 0. 0(:
We have |x (D) =
l (x)dx. We a s s u m e t h a t B w i l l n o t f a i l , a s l o n g a s
n
(
m
< p„ w h e r e p
c
0c
c
i s a p a r a m e t e r of t h e m o d e l . T h i s parameter indirectly
accounts for percolation effects, since when p increases, the domain D (where the breakage is expected to occur) progressively invades the specimen B . c
4.1. Critical
volume fraction
of defects
T h i s assumption can be used for non brittle damaging materials (neglecting the redistribution of the stresses during the process, or the growth of microcracks initiated on the sites with low critical stress) or for ductile fracture with cavities growing from inclusions. I n our probabilistic approach, o (x) is assumed to be a realization of a random function, and therefore l ( x ) is the indicator function of the random set D. We have c
0c
(43) Solving E q . (43) as a function of the probabilistic properties of a ( x ) and of the geometry of B (and of a(x)) requires finding the solution of a change of support problem. Since no general solution of E q . (43) is known, appropriate models or simplifications must be used, as is proposed below. It is easy to calculate the average and the variance of the random variable (x (D), depending on o (x). We have c
n
c
266
E[u (D)] = n
[ F(a(x),x)dx
(44)
where F(a(x),x) = P{a (x) < a(x)). E v e n when a is constant, the distribution may depend on x in B for a nonstationary random function o (x)). T h e geometrical covariogram of D, K ( h ) , is given by : c
c
0
j
K (h) = a
I
E [ l ( x ) l ( x + h)]dx= 0 c
0 c
F ( a ( x ) , a ( x + h))dx
(45)
2
JBRBh -/BnBh where F ( a ( x ) , o(x + h) = P{o (x) < a(x), o (x + h) < o(x + h)] is the bivariate distribution of o . T h e variance D [u. (B)] is obtained as : 2
c
c
2
c
n
-,2
D [n (B)] = f 2
K (h)dh -
n
(46)
[ F(a(x))dx
G
F r o m now on, we consider large specimens B and random functions o (x) for which the Central L i m i t Theorem can be applied. Asymptotically, u ( D ) is a Gaussian variable with average and expectation given by E q s . (44) and (46). Therefore, we get: P„-m 1 f P{no failure of B } = -7= e x p ( - y )dy (47) /"J — where m = E [ u ( D ) ] / u ( B ) and D = D [ t ( D ) ] / i ( B ) . c
n
&
2
2
n
2
n
2
(
n
(
n
I n the case of a stationary random function o (x), and a u n i f o r m s t r e s s f i e l d c
a over B , we get m = F ( a ) and D
2
= m ( l - m) A ( a ) / u ( B )
(48)
n
E q . (48) can be applied when the integral range A( o) of the random set, deduced from a ( x ) < o is different from 0 and remains finite. We have : c
2
f A(o)
F (a,a,h) - F (a) 2
" J„. W-FW')
DH
«
w i t h , F ( a , a, h) = P{a (x) < a, o (x + h) < a]. 2
c
c
We can also use the following empirical expressions D
2
29
= F ( o ) ( l - F(o))B(o)/1 |x (B) | » n
2
for the variance D : (50)
with a < 1 for A(o) = 0 and a > 1 for A(o) = oo . For the general mosaic model introduced above (not necessarily based on a Voronoi tesselation), in E q . (49), A(a) = A , and does not depend on a. We have 3
267 A
= J r(h) dh = 4jt j
3
2
h r(h) dh
rJ
for
an
isotropic
medium,
with
r(h) = K ( h ) / K ( 0 ) , K ( h ) being the geometrical covariogram of a random class A ' o f the tesselation (this can be measured by image analysis on polished sections of the material ): K(h) = V ( A ' f j '- )F r o m the law given by E q . (47), general properties of the fracture probabilities can be easily deduced : - For a given stress field o(x), P{fracture} decreases with the threshold p ; we have P{fracture} = 1/2 for m = p . 16
A
h
c
c
- For a uniform stress field a, the median macroscopic strength o is obtained from F ( o ) = p . T h e r e is therefore no size effect for o i n this model, which can be checked experimentally. T h e median o increases with p , while the variance D ( o ) decreases with V ( B ) . When V(B)-> «>, the fracture strength o converges to the constant a = F ( p ) : a deterministic fracture behaviour results on a large scale, towards a fracture strength depending both on F ( o ) and on p . M
M
c
M
M
c
2
R
R
_ 1
R
c
c
F o r t h e s t r e s s f i e l d g e n e r a t e d a t t h e c r a c k t i p of an elastic—plastic material, and with the simplifications introduced above, the volume V where fracture can occur by cleavage is limited to the plastic zone. We have:
where L is the length of the crack front, a the crack length, and a the area of the section of the plastic zone for a ( ^ ) = 1. T h e convergence towards the normal distribution is insured for a very large plastic zone, which involves An < Y where An is the integral range (assumed finite) obtained for the threshold o . T h i s corresponds to large stresses fulfilling: 2
0
When this conditions are satisfied, the part of the probability of fracture corresponding to the larger stresses is obtained as:
(51) and mo = F ( X o ) . - F o r p s mo , E q . (51) is not valid any more. I n that case, no fracture of the y
c
material can occur for any bounded stress a. - F o r p < m , the material breaks almost surely when o tends to infinity. c
268 4.2. Critical
density of
defects
T h e previous models cannot be applied to a m a t e r i a l c o n t a i n i n g defects w i t h a zero measure (points, t h i n microcracks, etc.). A n a l t e r n a t i v e c r i t e r i o n is a critical n u m b e r density, 0 . For a given applied stress field o ( x ) , N ( o ) is t h e r a n d o m n u m b e r o f defects w h e r e o ( x ) < o(x). We assume t h a t t h e f r a c t u r e w i l l occur w h e n N ( o ) > n , w i t h n = 0 u ( B ) ( n increases w i t h t h e c r i t i c a l density 0 , and w i t h the v o l u m e o f B ) . For a Boolean R a n d o m F u n c t i o n (Boolean V a r i e t y w i t h k = 0), we allow the superposition o f defects and N ( o ) follows a Poisson d i s t r i b u t i o n w i t h parameter C
c
C
f 6' =
n
C
T
c
^ H . ( o - a ) ) 0(dt) = n 0, z
t
m
Jo - W h e n n < 1 (for specimens B w i t h a s m a l l size), we recover t h e weakest l i n k c r i t e r i o n , and P{no failure of B } = exp(— 0') = p . — W h e n n > 1, we g e t : 0
P{no failure of B } = p
0
+ p .. + p r
m
+ ... + p „ = 1 - F ( 0 , ) n
(52)
I n the case o f p o i n t defects a n d a u n i f o r m stress field over B , we have 0' = 0V(B) ch(a) (where a)(a) is an increasing positive f u n c t i o n o f a, as earlier) a
and 9j = Q-
(0 n) I n E q . (52), p
r a
= P(N(o) = m) =
ra
±j
exp - nQ
v
W h e n o>(o) is derivable,
the d i s t r i b u t i o n F ( 0 i ) admits a density f ( 0 i ) and a mode n
- — i — - for a such
n
v e /
t h a t f-
(n — 1)!
(J>(a) = 1 (where e = exp 1). T h i s mode is u n i q u e w h e n 4>(a) is strictly
"c
monotonous. W h e n increasing t h e size o f B such t h a t n—• co, t h e p d f f ( o ) converges t o w a r d a D i r a c d i s t r i b u t i o n : t h e fracture s t r e n g t h o f t h e m a t e r i a l n
a
becomes a constant value o , such t h a t R
a>(o ) = 1. T h i s b e h a v i o u r is very R
s i m i l a r t o the previous model, based o n a c r i t i c a l v o l u m e f u n c t i o n . I t can also be s h o w n that, u n l i k e the weakest l i n k model, t h e large specimens are less sensitive to the most severe defects ( w i t h l o w o ) . T h i s type o f approach was used recently i n the case o f t w o d i m e n s i o n a l woven composites : the c r i t i c a l density model was used i n the i d e n t i f i c a t i o n o f defects on y a r n s m a d e o f a SiC/SiC composite, f r o m m e c h a n i c a l tests, a n d for the s i m u l a t i o n of damage by f i n i t e element calculations. I t appears t h a t the 7
c
15
269 tested specimens (with the two lengths 50 m m and 180 mmm) present a damage threshold during tensile tests. T h i s damage threshold is practically independent on the size, so that we used a critical damage density criterion to estimate the density of defects from these tests. T h e density was chosen according to a Weibull model with o = 0 and m = 2.45 in E q . (8). T h e agreement of this model with data is illustrated i n Figure 12 . 0
1
3i
0.9 0.8 0.7
t b
± x 0.6 a 0.5
CD
§ 0.4 Q_ 0.3 0.2 0.1 0I 0 0
120
140
50 exp
160
a
180 200 220 STRESS(MPa)
50 calc
180 exp
240
x
260
280
300
180 calc
Figure 12 : Comparison between the theoretical and the experimental cumulative distribution functions of the damage threshold observed on SiC/SiC yarns for two lengths (50 and 180 mm) (from ref 15).
5. F r a c t u r e s t a t i s t i c s m o d e l w i t h a c r a c k a r r e s t c r i t e r i o n I n this part, microstructural information along the crack path is used to estimate the probability of fracture of random media, as a function of microgeometrical characteristics, namely the spatial distribution of the specific fracture energy Y ( X ) . T h e approach is limited to crack propagation in two dimensional media. Some specific random media are considered (the Poisson
270 mosaic, and the Boolean mosaic random functions). Various loading conditions producing stable or unstable crack propagations, and the case of crack initiation from defects, followed by crack propagation, are studied. T h e models predict the probability laws of mechanical properties like strength, toughness (for crack initiation and arrest) and microcrack length. Indications on scale effects are provided. These results differ from the weakest link and the critical damage approaches developed above. 5.1. Crack propagation and the Griffith's criterion for two-dimensional
random
media The aim of this part is to calculate for some t w o - d i m e n s i o n a l random media the probability of fracture by application of the Griffith's criterion concerning the crack arrest during its propagation. Brittle random media, with a homogeneous constitutive law (namely elastic with Youngs' modulus E ) , but with a random fracture energy T(x) are considered. A s i n previous w o r k s , it is assumed that a possible fracture path P(s,d) connecting the source s to a destination d must satisfy, for every point x on the path: 8 - 1 0
3 0 - 3 2
2T(x) < G(x)
(53)
T h e energy release rate G(x), which is the elastic energy stored i n the loaded specimen depends on the location of the crack front x, on the loading conditions and also on the overall crack path from s to x. Given(53) a candidate crack path P, it must satisfy: V {2r(x) - G(x) ; x e P } < 0
(54)
where V is the supremum value of the expression i n brackets. For a given random function T(x), the probability of fracture must be calculated from E q . (54). T h i s is a very difficult task i n general, since on a microscopic scale, the crack path has a complex shape, depending on the microstructure. A s a result, the function T(x) i n E q . (54) is a realization of a random function correlated to T(x). Simplifications are introduced, and furthermore specific models are used to give closed form results, as shown below. T h e potential crack paths P can be assumed 30 31 32 to be realizations of a diffusion stochastic process independent on This involves crack paths which do not depend locally on the underlying microstructure. This last one is indirectly reflected by two parameters: a crack diffusion coefficient D and a coeeficient of smoothing. Following a n approach developed elsewhere , using a minimal fracture energy assumption, the crack paths P would be the shortest paths (or geodesic paths) on the field T(x), that can be determined by image a n a l y s i s - . Considering G(x) obtained for a straight
r--.
8
35
36
271 propagation of the crack, it is easy to calculate the criterion given by E q . (54) on simulations and to estimate a fracture probability. However, i n the present paper, we propose how to calculate i n a closed form the probability of extension of linear cracks along a segment, according to a crack advance i n mode I . T h e following crack configurations and loading conditions are examined, as shown i n Figures 13 , 14 , 15 .
t t I"
G(a)
a)
Figure 13 : Surface crack (initial length an) in a specimen with width (a+b), and variation of the energy release rate G(a), as a function of the crack advance, for two loading configurations: unstable crack propagation in a) and stable crack propagation in b). (from ref 9)
5.1.1. Surface
crack
A surface crack with initial length an i n a finite medium can propagate along a ligament with length b under two loading conditions (Figure 13 ): (i) a uniform stress a orthogonal to the crack is applied at infinity; i n that case, the energy release rate G(x) increases along the crack path, resulting i n an
272 unstable crack propagation for a homogeneous medium. For short cracks (ao + x < b/2), it can be approximated by the semi—infinite medium (up to a factor 1.25): G ( a + x) = Jt o ( a 2
0
+ x) / E
0
(55)
(ii) a concentrated load (also noted a for convenience) is applied at the mouth of the crack; for this configuration, G(x) decreases while the crack propagates, resulting i n a stable crack propagation which may be stopped; with the same approximation as for E q . (55), we have: G(a
0
+ x) = k o
2
/ [ E ( a + x)] 0
(56)
with k = 2.15 5.1.2 Internal
crack
A n internal crack with length 2a inside an infinite medium is uniformly loaded at infinity by a stress a; it can undergo a n unstable propagation from its two ends (Figure 14 ), with: G ( a + x) =
ii
0
t t 1°
o
2
( a + x) / E 0
(57)
»G(a)
I I I Ji G
Figure 14 : Internal crack (with initial length an)and variation of the energy release rate G(a), as a function of the crack advance. The other branch of the curve is obtained by symmetry (from ref 9).
5.1.3. Ring
test and crack
arrest
I n order to study the dynamic fracture of materials and the crack arrest properties, a ring test was developed - . For this type of specimen, submitted to 33
34
273 a diametral load noted a, the stress intensity factor follows a parabolic variation along the crack path. I n the present approach, the dynamic fracture properties are not accounted for and calculations are made in a static case for a parabolic shape of G(ao +x) according to Figure 15 , where A R is the difference between the two radii of the ring: G(a
0
+ x) = a
2
3
0
= h(a with h = a
2
2
( a + x)(k - ( a + x)) / ( E ( A R ) / ) 0
0
+ x)(k - ( a + x)) 0
/ (E(AR)3/2)
33
34
Figure 15 : Ring test configuration ' and modelled variation of the energy release rate G(a), as a function of the crack advance (with initial length ao): unstable propagation, followed by a stable crack propagation and arrest (from ref 9).
(58)
274 T h e m a x i m a l value o f G along t h e crack p a t h is g i v e n b y G
2
m a x
= h k / 4.
The
factor k is a p p r o x i m a t e l y equal to 3/4. 5.2. Types of Probability Distributions
obtained from the models
For a given geometry a n d r a n d o m m e d i u m , t h e p r o b a b i l i t y P(a,b,o) o f s t r a i g h t propagation o f a crack f r o m l e n g t h a to l e n g t h ( a + b ) w i l l be calculated as a f u n c t i o n o f a, b, a n d o. I t can be i n t e r p r e t e d i n various ways, according to the variable o f interest. (i) For a f i n i t e size specimen of w i d t h ( a + b ) , P(a,b,o) represents t h e p r o b a b i l i t y of fracture after i n i t i a t i o n f r o m a, u n d e r t h e considered l o a d i n g c o n d i t i o n . A rigorous approach w o u l d r e q u i r e i n t r o d u c i n g functions G(x) obtained for finite size specimens, as available i n t h e f r a c t u r e mechanics literature. For s i m p l i f i c a t i o n , we use here t h e functions v a l i d for i n f i n i t e or semi—infinite media. L e t o , a n d G be t h e overall f r a c t u r e stress a n d toughness o f t h e cracked specimen, as measured b y mechanical tests of f r a c t u r e f r o m precracked samples, w h i l e ^ is t h e c r i t i c a l l e n g t h of a microcrack. T h e i r p r o b a b i l i t y laws are related to P(a,b,a) by: R
c
P(a,b,o) = P{fracture}
= P K < °] = P { G < G(a)} C
= P { a < a}
(59)
c
(ii) For a f i n i t e size specimen o f w i d t h larger t h a n ( a + b ) , 1—P(a,b,a) represents the p r o b a b i l i t y of arrest of t h e crack before r e a c h i n g t h e l e n g t h ( a + b ) . L e t G and l be t h e arrest toughness o f t h e m e d i u m a n d t h e l e n g t h o f t h e crack at arrest. W h e n G increases w i t h t h e size o f the crack, t h e i r p r o b a b i l i t y laws are given by: a
a
P(a,b,ff) = P { f r a c t u r e } = P { G > G(a + b ) } a
= P { l > a + b}
(60)
a
I n E q . (60) t h e s y m b o l > for G is replaced b y < w h e n G decreases w i t h the crack l e n g t h . T h e various p r o b a b i l i t y d i s t r i b u t i o n s obtained f r o m E q . (59) and Eq. (60) for a g i v e n r a n d o m f u n c t i o n model T(x) are r e l a t e d i n a coherent way. I n practice, they are v e r y useful for t e s t i n g t h e m o d e l f r o m e x p e r i m e n t a l data obtained at different scales: mechanical data ( o , G , a*, G ) , as w e l l as crack size d i s t r i b u t i o n observed for given l o a d i n g conditions (ac, l ) . a
R
c
a
a
(iii) W h e n changing t h e size o f t h e specimens, size effects can be p r e d i c t e d o n the p r o b a b i l i t y d i s t r i b u t i o n o f t h e mechanical properties a n d o n t h e i r m o m e n t s , such
275 as their average value. For large size specimens, asymptotic results can provide different behaviors, as illustrated i n the next parts for specific random function models. (iv) A combination of crack initiation and propagation (or arrest) can be modelled in a simple way as follows: consider i n the random medium critical defects located according to a random point process with intensity a>(a )(the average number of defects per unit area, with critical stress o less than a). For the defects where a < a, a microcrack with length ao is initiated, ao being a random variable, or more simply a constant material property. For a low intensity of defects, their crack paths during propagation are uncorrelated (with respect to T(x)), and the complex calculation of G resulting from a network of microcracks can be avoided; therefore microcrack interactions are neglected. I n addition, for large specimens it can be assumed that the probability of propagation of ao to the length (ao+b) is independent on the location x of the defect. L e t G ( s ) be the probability generating function of the discrete probability p for the random number of critical defects N(o) contained i n a rectangular specimen with sides b and d. Given N(o) = n, it comes: c
c
a
n
P{Non fracture / N = n] = ( l - P ( a , b , o ) )
n
(61)
0
and therefore: P { N o n fracture} = G ( l - P ( a , b , o ) ) Q
(62)
0
When accounting for the probability distribution of the independent locations x of the defects, P(ao,b,a) i n E q . (62) must be replaced by EP(ao,b(x),o), obtained as an average of P(ao,b(x),o) over these locations. A n important particular case is given by a Poisson point process (for which when N ( a ) = n, the defects are independently and uniformly distributed i n the material) with intensity 4>(a), as used i n weakest link models (see section 3.3.2) with generating function: G ( s ) = exp{b d c>(a) (s - D }
(63)
a
For this model E q . (62) becomes: P { N o n fracture} = exp{ - b d (b(o) P ( a , b, a)}
(64)
0
Similar expressions can be obtained for surface cracks, when restricting the nucleation on defects located on stripes along the edges of the specimen. Illustrations will be given i n the next two subsections i n the case of two models of random functions for T(x): the Poisson mosaic, and the Boolean functions.
random
276 5.3. Probability
of fracture
and scale effects for the Poisson
Mosaic
T h e Poisson mosaic is a p a r t i c u l a r model, w h i c h m i g h t s i m u l a t e a r a n d o m polycristal w i t h local changes o f f r a c t u r e energy. D i f f e r i n g f r o m t h e V o r o n o i mosaic used i n section 3.4.2, i t is b u i l t i n t w o steps: - a Poisson tessellation w i t h parameter X. d e l i m i t s the g r a i n boundaries; t h e y are made o f Poisson lines i n t h e plane for a t w o - d i m e n s i o n a l m e d i u m ; - for grains o f t h e tessellation, the fracture energy y are independent realizations o f a r a n d o m variable T, w i t h t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n F ( y ) = P { r < y } ; t h i s is also t h e d i s t r i b u t i o n f u n c t i o n o f t h e r a n d o m f u n c t i o n T(x) b u i l t b y t h e r a n d o m mosaic. T h e p r o b a b i l i t y o f fracture o f a specimen loaded according t o Eqs. (55), (56), (57), (58) is derived f r o m the fact t h a t , for the Poisson mosaic, t h e n u m b e r of grains h i t b y a segment of l e n g t h 1 follows a Poisson d i s t r i b u t i o n w i t h parameter
5.3.1. Surface
crack
I n t h i s section, i t is assumed t h a t a surface crack w i t h i n i t i a l l e n g t h 2a propagates i n the r a n d o m m e d i u m u n t i l the l e n g t h 2 (a + b ) . i) Unstable crack propagation For the unstable crack propagation g i v e n i n E q . (55), or for a n y l o a d i n g where G increases w i t h x, t h e p r o b a b i l i t y o f fracture P ( a + b ) is g i v e n by: 2(a + b)
(1 - F ( G ( u ) / 2 ) ) d u
(65)
2a
The p r o b a b i l i t y o f fracture of the specimen increases (and converges t o w a r d 1), w h e n t h e crack l e n g t h 2a or t h e applied stress o increases. I t also increases for lower X corresponding to a coarser m i c r o s t r u c t u r e : w i t h t h i s m o d e l , s m a l l grains i m p r o v e t h e a b i l i t y to resist the crack g r o w t h , as a r e s u l t o f a h i g h e r p r o b a b i l i t y to meet grains w i t h a large fracture energy y along t h e same crack p a t h . Considering n o w samples w i t h a s i m i l a r geometry ( w i t h a constant r a t i o a/b), scale effects depending o n the d i s t r i b u t i o n F( y) are observed. F i r s t , i f t h e range of the d i s t r i b u t i o n is f i n i t e ( F ( y ) = l f o r y > Y c ) , t h e p r o b a b i l i t y o f f r a c t u r e becomes equal to 1 beyond t h e c r i t i c a l fracture l e n g t h 2a<; g i v e n b y 2 a = 4y E / ( j t o ) . I f the d i s t r i b u t i o n F ( y ) possesses a t a i l such as 1 - F(y) = y~ w h e n y becomes i n f i n i t e , the scale effects s t r o n g l y depend o n t h e positive coefficient a: * i f a = 1, for large specimens, t h e asymptotic p r o b a b i l i t y o f f r a c t u r e becomes independent on t h e i r size (there is no scale effect): 2
c
a
c
277 2
P ( a + b) = * if a
with y = a/(a+b) and c = E/(2jr.a )
(66)
1, the large scale behavior of the probability of fracture becomes: a
1
a
a
1
P ( a + b) = e x p [ - X . ( 2 c ) a - ( l - y " ) / ( a - 1)]
(67)
W h e n a < 1, the growth of F ( y ) towards 1 is so slow that the crack is stopped with a probability 1 by grains with fracture energy as large as required for the crack arrest condition (P(a+b) converges to 0 for increasing sizes). When a > 1, P(a+b) converges to 1 for increasing sizes, as for a distribution with a finite range. A s previously mentioned, from the probability P(a+b), other statistics can be derived from the knowledge of F ( y ) ; to illustrate this, we just consider here asymptotic results for a power law tail of the distribution: - Replacing c by its expression, the fracture stress distribution is obtained. Scale effects concerning the change of the median strength o with the size of the specimen are the following: M
* if a = 1, er = J— Logy, and there is no scale effect; * if a ^ 1, a = L " ^ | 1 — y | / , where L is the size of the specimen ; therefore o increases with d for a < 1 and o decreases with d for a > 1. - T h e statistics for the toughness G is obtained from E q s (59), (66), (67), and from G = a/c: M
a - 1
1
( 2 a )
M
M
M
c
c
* i f a = 1: P{G
C
( X a
< G} = y
/
G )
(68)
The median toughness is proportional to Xa. •if a *
1: P{G
C
< G } = exp -
( f l
_
(69)
- T h e probability distribution for the critical length ac (for 2a < L ) is: * if a = 1: P{a * if a *
c
< a} = ( 2 a / L )
X c
(70)
1:
P{a
a
c
< a } = exp [ - X(2c) ( L / 2 )
1 _ a
1
( l - ( 2 a / L ) - « ) / ( a - 1)]
T h e median critical length is proportional to the specimen size L . - T h e probability distribution for the arrest length l (with l > a) is: a
* if a = 1:
a
(71)
278 P{l
a
> 1} = (a/1)**
(72)
l follows a Pareto distribution, and its median value is proportional to a; a
*if a * 1 : P{l
> 1} = exp [ - X ( 2 c ) a a
a
1
a
1
- ^ - ( a / l ) " ) / ( a - 1)]
(73)
a
T h e median length at arrest is proportional to a for a > 1 and to a for a > 1. — T h e probability distribution for the toughness at arrest G is obtained from the expression G = (a+b)/c, and from Go = a/c (with G > Go): a
a
* if a = 1 : P{G
a
> G} = (G /G)
t e
(74)
0
G follows a Pareto distribution, and its median value is proportional to a; * if a *t 1 : a
P{G
> G } = exp [ - X ( 2 c ) a - ( l - G / G ) " ) / ( a - 1)] a
a
1
a
a
1
0
(75)
T h e median value of G is proportional to a. ii)Stable crack propagation For the stable crack propagation given in E q . (56), or for any loading where G decreases with x, P(a+b) becomes: a
,2(a+b)
-AJ
(1 - F ( G ( u ) / 2 ) ) du
(76)
P(a+b) converges towards 0, whatever the distribution F, contrarly to the other loading condition. T h e crack is stopped almost surely for specimens with increasing sizes. 5.3.2. Internal
crack
For the loading given by E q . (57) the probability of fracture P(a+b) corresponding to the configuration of Figure 14 is given by: ,2(a + b)
P(a + b) = P ( a ) exp 0
2X\
(1 - F ( G ( u ) / 2 ) ) du
(77)
'2a
I n E q . (77), P ( a ) is the probability for the initial crack (2a) to start from its both ends. I t is given by: 0
P„(a) = r(2a) F ( G ( 2 a ) / 2 ) + (1 - r(2a)) F ( G ( 2 a ) / 2 ) r(2a) = e x p ( - 2Xa)
2
(73)
279 T h e first t e r m i n E q . (78) expresses that the two ends of the crack are i n the same cell of the Poisson tesselation and the second term that they belong to two different cells. U p to Pn(a), the fracture probability given by E q . (77) is very similar to E q . (65). Therefore the above derived statistics and the corresponding scale effects, given for large specimens, still apply for the internal crack problem, provided the factor X. is replaced by 2X.. 5.3.3. Parabolic
loading
For the loading corresponding to E q . (58) and Figure 15 , the probability of fracture P ( a + b ) is given by (with G i given by the smallest value of G along the crack path): m
P ( a + b) = F ( G
m i n
n
/ 2 ) exp| - X j
(1 - F ( G ( u ) / 2 ) ) du
(79)
A s a particular case, it is interesting to consider as before a distribution F ( y ) with a power law tail. Explicit results are easily derived when a = 1, for which E q . (79) becomes, with h given i n E q . (58):
~ l
k P
(
a
+
b) =
[a + b
(a+ b )
(2X/kh)
(80)
k —a
For this model, there is no scale effect, as for the previous cases. U s i n g E q . (58) the probability laws of l and of G are given by: a
a
P{G
a
P{G
a
(81)
> 1} = I ^ S a
with a < 1 < k (2X/kh)
2
with l = k 5.3.4. Crack
2
> G} =
- 4 G / h and G < G initiation
and
m a x
k - ak + 1
(82)
.
propagation
Starting from E q . (64) and using the results given above, the probability of fracture may be derived for various situations. It is interesting to examine surface and internal crack initiation. I t turns out that, with the simplifications made to obtain E q . (64), the observed scale effects are the same i n the two cases so that the presentation will be limited to surface crack initiation by means of E q s (65), (66), (67) for a distribution F ( y ) with a power law tail.
280 * if a = 1 :
{ N o n fracture} = exp j - a da>(o) 0
-
° g
^
(83)
Scale effects are obtained for similar specimens, when b and d are replaced by kb and by kd. - When
= 1, there is no scale effect. 1
no - When
> 1, P { N o n fracture} converges to 1 for large scales. This KO
2
corresponds to low stresses. - When
< 1, P { N o n fracture} converges to 0 for large scales. This JtO
2
corresponds to large stresses. * if a
*
1 : P { N o n fracture} = e x p { - a drh(o) P ( a , b , a ) } 0
(84)
0
a-n
E
P ( a , b, a) = exp 0
,1-a
a
(a - 1)
n
+ b
- W h e n a < 1, P {Non fracture} converges to 1 for large scales. — When a > 1, P { N o n fracture} converges to 0 for large scales. 5.4. Probability
of Fracture
and scale effects for the Boolean
Mosaic
T h e Boolean random functions were used earlier i n this chapter when applying the weakest link criterion. A different construction is used i n the present section. A particular class is given by the Boolean mosaic, which is obtained as follows: — a material with a constant fracture energy y is considered; for simplification i n the notations, assume that Yn 0; it is equivalent to examining the case of cracks with lengths larger than 2an, that should be propagating i n the homogeneous material according to the criteria given by E q s . (53) (54). — on every point of a Poisson point process (with intensity 8(u)), is implanted a random function T'(u). T h e value T(x) of the fracture energy at point x is given by the supremum over all the "primary" random functions V covering x. For the general model, any dependence of T' on u is allowed. I n the present case, a constant fracture energy (y = u > Yn) inside realizations of a random grain X ' i s considered. T h e measure 0 is such that f 0(u)du remains finite. A n example of simulation is shown on Figure 5 . 0
=
281 T h e m a i n morphological difference with the Poisson mosaic is that, if the previous model is well suited for simulating a polycristal, the Boolean random function w i t h convex grains X ' is a good simulation of a matrix with a constant fracture energy y containing reinforcing inclusions (as opposite to defects for the weakest link model) with a larger fracture energy y. T h e probability of fracture deduced from the criterion (54) is obtained for a given crack path by: 0
P{fracture} = exp [ - | i ]
(85)
0
I n E q . (85), the coefficient u depends on the loading conditions through G(x), on the measure 0(y), and on the random set X', that is assumed to be a convex set. This equation is obtained as a particular case of the properties of the Boolean random functions. I t is developed below for the different loading conditions. As for the Poisson mosaic, it is possible to work out from E q . (85) the probability distributions of various mechanical properties (toughness, length of crack at arrest, and so on). T h i s is not presented here. 0
5.4.1. Surface
crack
i) Unstable crack propagation I n the case of the unstable crack propagation corresponding to E q . (55) and to Figure 13 the following expressions of |x are obtained for the mentioned family of "primary" random functions F ' : 0
*
u
n
= 0
*
u
0
= A ( X ' © 2b) = A ( X ' ) + L(X')2b/jr
*
n
0
= A ( X ' ® 21(y)) = A ( X ' ) + + L(X')21(y)/jt
with
if
2y < G(a)
(86)
2
l(y) = (2y - G ( a ) ) E / ( j t o ) ,
if
if
2y > G ( a + b)
G(a) < 2y < G ( a + b)
I n E q . (86), A(X') and L ( X ' ) are the average area and perimeter of the random convex set X ' ; the symbol © stands for the dilation operation defined i n mathematical morphology as before. It becomes: 16
,+00
u
0
= A ( X ' © 2b)
,G(a+b)/2
6(u)du + JG(a + b)/2
A ( X ' © 21(u)) 9(u)du
(87)
JG(a)/2
For a given measure 6, the probability of fracture is given exactly by a combination of E q s . ( 8 5 ) - ( 8 7 ) . T h e probability of fracture converges towards 1 when increasing separately the crack length 2a or the applied stress a.
282
The effect of the size of the inclusions is the following: the probability of fracture decreases with the size of the reinforcing inclusions X ' . T h i s is due to the fact that, with this size, the overall area fraction of higher strength material increases. I f 0(y) = O w h e n y s Yc, the probability of fracture reaches 1 for large specimens where the crack length 2a is beyond the critical fracture length 2ac 2
given by 2 a = 4 y E / ( n a ) . For other measures, the asymptotic behavior of large c
c
size specimens depends on the tail 9(y) = 6 y
- a
of the measure 8 for large y. T h e case of
with a > l is presented for illustration. With this intensity, the
cumulative distribution function of the random function T is: F(y) =
P{r(x)
(88)
1
< y) =
exp^-^y -"
For large values of y, a power law tail of the distribution is obtained, since: l - F ( y ) = ^
T
T h i s is equivalent to the Poisson mosaic model with similar distributions, leading to Eqs. (66), (67). * I f ct=2, we obtain n
0
= 40c[A(X')/a - 2 L ( X ' ) Log(y)/jt]. I t results:
P(a + b) = 8 e c L C X ' ) A y
e x p
r _ e (A(X')/a] 4
(89)
c
with y and c given i n E q . (66) to which E q . (89) can be compared; it corresponds to a similar situation for a different model: the fracture probability
slightly
increases with the size of similar specimens, and converges towards a constant depending only on the geometry (through y) and on the microstructure (through 0 and L ( X ' ) ) . For this specific intensity 0, there is no size effect on the fracture statistics. * If a * 2 :
=
4
9|f I
T T ^ - T I
A(X') + 2 L ( X ' ) a/rt^'
2
-
1
(a -
1)1
(90)
When a < 2, u diverges for increasing sizes of the specimen, so that the probability of fracture converges to 0. When a > 2, \i converges to 0, and the probability of fracture of large specimens converges to 1. T h i s is consistent with the results obtained for a Poisson mosaic and a similar distribution function (after replacement of a in the previous case by c t - 1 i n the present case). T h e same scale effects for the median strength a as before are observed, ii) Stable crack propagation 0
0
For the stable crack propagation given by E q . (56), the coefficient m i n E q . (85) is obtained as follows:
283
*
u
0
= 0
if
2y < G ( a + b)
*
n
0
= A ( X ' © 2b) = A ( X ' ) + L(X')2b/jr,
*
M-O = A ( X ' © 2(b - 1(Y))) = A ( X ' ) + L(X')2(b 2
with
1(Y) = k o / ( 2 E Y ) - a
if
if
2y > G(a) 1(Y))/JI
G ( a + b) < 2y < G ( a + b)
I n the present case, E q . (87) becomes, with G(a) given by E q . (56): /•+»
\i
f
= A ( X ' © 2b)
0
G(a)/2
9(u)du + JG(a)/2
A ( X ' © 2(b - l(u)))0(u)du (91) JG(a+b)/2
It is easy to deduce from E q . (91) that the coefficient \i diverges with the size of the specimen, so that the probability of fracture becomes 0 for large samples, as for the Poisson mosaic model. 0
5.4.2. Internal
crack
For the loading given by E q . (57) corresponding to the configuration of Figure 14,
the following
expressions
of
(x are obtained, when considering the 0
approximation of cracks much larger than the size of the inclusions X ' : *
u
*
\i
*
\i
0
= 0
if
2Y
< G(a)
0
= 2 A ( X ' © b) = 2A(X') + L(X')2b/jt
Q
= 2 A ( X ' © 1(Y)) = 2A(X') + L ( X ' ) 2 1 ( ) / n
with
if
2y > G ( a + b)
Y
l(y) = ( 2
2
Y
- G(a))E/(jto ),
if
G(a) < 2y < G ( a + b)
Therefore n is given by : 0
, + oo
\i
0
= 2 A ( X ' © b)
,G(a+b)/2
9(u)du + 2 •'G(a+b)/2
A ( X ' © l(u))9(u)du
(92)
JG(a)/2
Since E q . (92) is very similar to E q . (87) obtained for the unstable propagation of a surface crack, the previously obtained results concerning scale effects are observed again for this internal crack configuration. 5.4.3. Parabolic
loading
For the loading corresponding to E q . (58) and Figure 15 , (x is expressed as 0
follows,
with G
m
a
x
given i n E q . (58) and G
m
, = m i n (G(a), G(a+b)); the n
approximation of b m u c h larger than the size of the inclusions X ' is made.
284 *
= 0
H
if
2
Y
<
*
| i = A ( X ' © b) = A ( X ' ) + L ( X ' ) b / j t
*
n
if
0
0
2y >
G
mBX
= 2 A ( X ' ) + L(X')21(Y)/Jt. Gma
with 21(y) = b - 2
y *~
2 y
h
if G
2
= m a x ( G ( a ) , G ( a + b)) < 2
Y
< G
n
* For b < k - 2a, G ( a + b ) > G(a), and for G(a) < 2y < G ( a + b ) : (i
y
with 1(Y) = k / 2 - a -
= A(X') +
0
Gmax h
"
L(X')1(Y)A
2Y
* For b > k - 2a, G ( a + b ) < G(a), and for G ( a + b ) < 2y < G(a): M-o =
with 1(Y) = a + b -
-y
k/2
A(X') +
Gmax
L(X')KY)/^
27 h
Therefore j i is given by: 0
+ oo
|i
0
^. U m a x / Z
6(u)du +
= A ( X ' © b) J Gmax/2
A ( X ' © Ku)) 6(u)du J G , ^
G /2 2
f
I
I,
A ( X ' © Ku)) 0(u)du
(93)
Gmin
Specific results can be deduced from E q . (93) for particular measures 0(Y)5.4.4. Crack
initiation
and
propagation
Combining E q . (64) to the results given i n the previous subsections, the probability of fracture may be derived for surface and internal crack initiation. T h e case with 0 ( Y ) = 0
_
Y
is studied below for large scale effects.
i) Surface microcracks * I f a = 2, using E q . (89) enables us to draw the following conclusions: — When 8 0 c L ( X ' ) / n =1, there is no scale effect on the probability of fracture. — W h e n 80cL(X')/it > 1, P { Non fracture } converges to 1 for large scales. — When 8 0 c L ( X ' ) / n < 1, P { Non fracture } converges to 0 for large scales. * I f a * 2, using E q . (90) we get:
285 - W h e n a<2, \i diverges 1 for large scales.
for large specimens, and P{Nonfracture} converges to
0
- W h e n a > 2, (x converges to 0 and P{Nonfracture} converges to 0 for large scales. The asymptotic behavior of the probability of fracture is very similar to the Poisson mosaic case with the same fracture energy distribution, ii) Internal microcracks 0
The internal microcracks with length 2ao, are assumed to be much smaller than the reinforcing inclusions X'. Therefore a different approximation than in the case leading to E q . (92) is made, which gives: *
u
0
= 0
if
2
*
u
0
= A ( X ' © 2b) = A ( X ' ) + L(X')2b/jt
*
u
0
= A ( X ' © 21( )) = A ( X ' ) + L(X')21( )/Jt
Y
< G(a ) 0
Y
w i t h 1(Y) = 2Y E / ( J I O
if
2y > G ( a + b) 0
Y
2
) if G ( a ) 0
< 2
< G(a
Y
0
+ b)
* It comes for a = 2 (see E q . (89)): I-IQ
^
y
= 40c A ( X ' ) / a
0
+ L(X')/JI
a
0
2b - 2Log(y + b
-1
(94)
° = aTTb
For large sizes b, u diverges so that P(an,b,o) exponentially converges to 0 and P{Nonfracture} converges to 1 for large scales. T h i s is different from the macroscopic crack a and from the surface nucleation. * I f a * 2: 0
4c
a-l
1 a - l
r A(X')
yO-2
+ 2 L ( X •)/n
a
\hy ~
l
_ I
(95)
When a < 2, u diverges for large scales so that P(an,b,a) exponentially converges to 0 and P{Nonfracture} converges to 1. W h e n a > 2, u converges towards a finite value for large scales, so that P{Nonfracture} converges to 0. Again similar results as for the Poisson mosaic are obtained. 0
0
5.5.
Conclusions
It was possible to derive a theoretical probability of fracture of unstable or stable straight crack propagation, due to various loading configurations, in two
286 types of random media, namely the Poisson mosaic and Boolean random functions. Resulting size effects are similar for the two models, and are strongly dependent on the tail of the cumulative distribution function of the fracture energy y modelled by a power law with exponent a: for instance, for a large surface or internal crack, in the case of a very slow growth of F ( y ) ( a < 1), the probability for large specimens to encounter a locally large fracture energy is so important that the probability of fracture decreases to 0. W h e n a = 1, the probability of fracture of large specimens is a constant depending on the geometry and on the microstructure, but not on the size. W h e n a > 1, in any large specimen, a long crack is subject to propagation until fracture. Similar results are obtained for crack initiation and propagation on critical defects. T h e observed size effects may differ from the results of the weakest link model where the average strength of a material always decreases with the specimen size. Additional probability laws concerning the fracture strength, the toughness (at crack initiation or at crack arrest), and the size distributions of critical microcracks and stable microcracks were derived. These results can be used to interpret experimental data obtained on materials. 6. C o n c l u s i o n T h e models introduced above present some simpUfications coming from the basic assumptions recalled in section 2. T h e i r main advantages are the following: —Various morphologies of microstructural defects (inclusions in a matrix, polycrystals, fiber and strata,...), inducing correlations on various ranges, can be described and simulated. —Exact theoretical results, coherent at different scales, are available. —Depending on few parameters (2, 3 or more), they can be tested from data on various scales: on the macroscopic scale, by means of the experimental distributions obtained from mechanical tests on various specimens geometries; on a microscopic scale, by means of image analysis measurements after localization of the d e f e c t s ' . 21,22
23
- Various scaling laws are obtained, according to the chosen fracture criteria, or to the appropriate random structure models, reflecting the situations occurring with experimental data: the overall strength of specimens may decrease with their size (weakest link), may be size invariant (damage threshold), or may even increase (crack arrest). However, one must remain aware of the fact that different combinations of fracture criteria and microscopic models can result in the same size effects, as underlined by D. J e u l i n . Therefore it is unwise to draw 7
287 definite conclusions solely on size effects, without any indication on the microstructure and on the m i c r o - m e c h a n i s m s of fracture. —The models can be easily introduced i n post—processor calculations i n a finite element c o d e ' . N e w extensions combine the simulation of population of defects after their identification from the models and mechanical tests, and the simulation of the progression of damage i n composites by finite element calculation ; this is a way to account for stress redistribution i n microcracking processes, which is difficult to handle by analytical calculations. 2 1 , 2 2
2 3
15
T h e use of these models is not restricted to the simulation of critical fracture criteria, as proposed i n this paper. I n fact they are able to simulate other physical random media with a microstructure, including the distribution of multivariate or tensors properties, that can be used for other purposes such as homogenization calculations. 7. R e f e r e n c e s 1.
G . Matheron, Elements pour une theorie des milieux poreux (Masson, Paris,
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E . Sanchez
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I N S T N , 4 (Ed. de la Revue de Metallurgie, 1991)
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Keynote
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23
geodesic
Series on Advances in Mathematics for Applied Sciences Editorial
N . Bellomo
G. P. Galdl
Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy
Institute of Engineering University of Ferrara Via Scandiana 21 44100 Ferrara Italy
Editor-in-Charge
Editor-in-Charge
A. V. B o b y l e v
K. R. Rajagopal
Keldysh Institute of Appl. Math. Miusskay Sq. 4 Moscow 125047 Russia
Mech. Eng.ng Department University of Pittsburgh Pittsburgh, PA 15261 USA
C. M. D a f e r m o s
R. R u s s o
Lefschetz Center for Dynamical Systems Brown University Providence, RI 02912 USA
Dipartimento di Matematica Universita degli Studi Napoli II 81100 Caserta Italy
J. G. H e y w o o d
V. A. S o l o n n l k o v
Department of Mathematics University of British Columbia 6224 Agricultural Road Vancouver, BC V6T 1Y4 Canada
Institute of Academy of Sciences St. Petersburg Branch of V. A. Steklov Mathematical Fontanka 27 St. Petersburg Russia
S. Lenhart
F. G. T c h e r e m i s s i n e
Mathematics Department University of Tennessee Knoxville, TN 37996-1300 USA
Computing Centre of the Russian Academy of Sciences Vasilova 40 Moscow 117333 Russia
P. L. L i o n s
University Paris Xl-Dauphine Place du Marechal de Lattre de Tassigny Paris Cedex 16 France S. K a w a s h i m a
Department of Applied Sciences Faculty Eng.ng Kyushu University 36 Fukuoka 812 Japan B. Perthame
Laboratoire d'Analyse Numerique University Paris VI tour 55-65, 5ieme etage 4, place Jussieu 75252 Paris Cedex 5 France
J . C. W i l l e m s
Faculty Mathematics & Physics University of Groningen P. O. Box 800 9700 Av. Groningen Groningen The Netherlands
Series on Advances in Mathematics for Applied Sciences A i m s and
Scope
This Series reports on new developments in mathematical research relating to methods, qualitative and numerical analysis, mathematical modeling in the applied and the technological sciences. Contributions related to constitutive theories, fluid dynamics, kinetic and transport theories, solid mechanics, system theory and mathematical methods for the applications are welcomed. This Series includes books, lecture notes, proceedings, collections of research papers. Monograph collections on specialized topics of current interest are particularly encouraged. Both the proceedings and monograph collections will generally be edited by a Guest editor. High quality, novelty of the content and potential for the applications to modern problems in applied science will be the guidelines for the selection of the content of this series.
I n s t r u c t i o n s for
Authors
Submission of proposals should be addressed to the editors-in-charge or to any member of the editorial board. In the latter, the authors should also notify the proposal to one of the editors-in-charge. Acceptance of books and lecture notes will generally be based on the description of the general content and scope of the book or lecture notes as well as on sample of the parts judged to be more significantly by the authors. Acceptance of proceedings will be based on relevance of the topics and of the lecturers contributing to the volume. Acceptance of monograph collections will be based on relevance of the subject and of the authors contributing to the volume. Authors are urged, in order to avoid re-typing, not to begin the final preparation of the text until they received the publisher's guidelines. They will receive from World Scientific the instructions for preparing camera-ready manuscript.
